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On Group and Semigroup Algebras

Paula Murgel Veloso

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‘Would you tell me, please, which way I ought to go from here?’

‘That depends a good deal on where you want to get to,’ said

the Cat.

‘I don’t much care where–’ said Alice.

‘Then it doesn’t matter which way you go,’ said the Cat.

‘–so long as I get somewhere,’ Alice added as an explanation.

‘Oh, you’re sure to do that,’ said the Cat, ‘if you only walk

long enough.’

(Lewis Carroll, Alice’s Adventures in Wonderland)

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Agradecimentos

Ao Prof. Arnaldo Garcia, do IMPA, pela orienta¸c˜ao do meu mestrado e

do meu doutorado, e pelo incentivo.

Ao Prof. Guilherme Leal, da UFRJ, pela co-orienta¸c˜ao do meu doutorado,

por ter-me proposto o problema que resultou no segundo cap´ıtulo desta tese,

e pelo apoio constante.

Aos Profs. Eduardo Esteves, do IMPA, Jairo Gon¸calves, da USP, Pavel

Zalesski, a UNB, e Amilcar Pacheco, da UFRJ, membros da banca de defesa

de tese, pelos coment´arios e sugest˜oes.

To Prof. Eric Jespers, from Vrije Universiteit B russel, for having proposed

the problem presented in the third chapter of this thesis, for the elucidative

conversations, good advice and kind hospitality during my stay in Brussels.

A Prof. Luciane Quo os, da UFRJ, pelas discuss˜oes matem´aticas sempre

esclarecedoras e bem-humoradas, e pela colabora¸c˜ao no artigo no qual se

baseia o segunto cap´ıtulo desta tese.

To Dr. Ann Dooms, from Vrije Universiteit Brussel, for the nice and

fruitful discussions and friendly support, and for the collaboration on the

article in which the third chapter of this thesis is based.

Aos Profs. Paulo Henrique Viana (in memoriam), Carlos Tomei e Pe.

Paul Schweitzer, da PUC-Rio, por me mostrarem a beleza da Matem´atica e

pela amizade.

Aos professores do IMPA, em especial aos Profs. Karl-Otto St¨ohr, C´esar

Camacho, Paulo Sad e Manfredo do Carmo.

A todos os meus colegas do IMPA, principalmente aos meus amigos

de turma do mestrado e aos amigos da

Algebra Juscelino B ezerra, Cleber

Haubrichs, Juliana Coelho e Andr´e Contiero.

Aos professores e alunos grupo de

Algebra N˜ao-Comutativa da UFRJ.

To the colleagues, professors and friends from Vrije Universiteit Brussel,

especially Isabel Goﬀa, Julia Dony and Kris Janssen.

A todos os funcion´arios do IMPA, em especial `aqueles da Comiss˜ao de

Ensino, da xerox e da seguran¸ca.

Ao CNPq, pelo apoio ﬁnanceiro e pela oportunidade de passar um ano

na Vrije Universiteit Brussel, Bruxelas, B´elgica, no programa de doutorado-

sandu´ıche.

Aos amigos Leandro Pimentel, Lourena Rocha, Marcos Petr´ucio Caval-

cante, Cec´ılia Salgado, Jos´e Cal Neto, Ricardo Bello, Taissa Abdalla, Kak´a

Boa Morte, Yolande Lisb ona, Wanderley Pereira, Alexandre Toledo, Helder

Gatti, Raul Tanaka, Eri Lou Nogueira, Paula Avellar, S´ergio Leiros, Karla

Suite, Cristina Diaconu, Maria Agustina Cibran, pela amizade, pela com-

panhia em todos os momentos, por enriquecerem e iluminarem minha vida.

A minha fam´ılia, principalmente aos meus pais Paulo Augusto e Sheila

Regina, e `a minha irm˜a Fl´avia, pelo ambiente familiar sempre alegre e aco-

lhedor, pelos constantes carinho, incentivo, paciˆencia e exemplo.

Paula Murgel Veloso

Julho, 2006

Contents

Agradecimentos i

Introduction iv

1 Preliminaries 1

1.1 Group Ring Theory . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Basic Deﬁnitions . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 Some Results . . . . . . . . . . . . . . . . . . . . . . . 10

1.2 Semigroup Ring Theory . . . . . . . . . . . . . . . . . . . . . 14

1.2.1 Prerequisite: Semigroup Theory . . . . . . . . . . . . . 14

1.2.2 Basic Deﬁnitions and Some Important Results . . . . . 21

2 Central Idempotents of Group Algebras of Finite Nilpotent

Groups 25

2.1 Primitive Idempotents of Semisimple Group Algebras of a Fi-

nite Abelian Group . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 Primitive Central Idempotents of Complex Group Algebras of

a Finite Nilpotent Group . . . . . . . . . . . . . . . . . . . . . 31

2.3 Some Questions for Further Investigation . . . . . . . . . . . . 38

3 The Normalizer of a Finite Semigroup and Free Groups in

the Unit Group of an Integral Semigroup Ring 40

3.1 The Normalizer of a Semigroup . . . . . . . . . . . . . . . . . 41

3.1.1 Characterization of N(±S) and Some Results . . . . . 44

3.1.2 The Normalizer Problem for Semigroup Rings . . . . . 53

3.2 Free Groups generated by Bicyclic Units . . . . . . . . . . . . 54

3.3 Some Questions for Further Investigation . . . . . . . . . . . . 55

Bibliography 57

Index 60

iii

Introduction

In this work, we are interested in Group Ring Theory. Although some

issues in Semigroup Ring Theory are also presented, the main focus is group

rings.

In Chapter 1, the basic theoretical foundations in group and semigroup

rings are laid. These ideas will be used throughout the entire work.

We begin with a presentation of deﬁnitions and theorems from Ring The-

ory, establishing their particular instances in the context of group rings. We

deﬁne group rings; the augmentation mapping; involutions; the Jacobson

radical of a ring; idempotents; simple comp onents; the character table of a

group; trivial, bicyclic and unitary units; the upper central series, the FC

center and the hypercenter of a group. Important results from Ring and

Group Ring Theory are then recalled, such as Wedderburn-Malcev theorem,

the decomposition of a semisimple ring in a direct sum of ideals, Wedderburn-

Artin theorem, Maschke theorem. For the sake of completeness and because

this may be elucidating in the sequel, some results are presented with their

proofs (Perlis-Walker theorem, the character method for obtaining primitive

central idempotents in group algebras over complex ﬁelds, Berman-Higman

lemma and its corollaries).

As a prerequisite to the study of semigroup rings, we brieﬂy present

fundamental concepts from Semigroup Theory. We deﬁne semigroups and

special kinds of semigroups (regular, inverse, completely 0-simple, Brandt);

idempotents; principal factors and principal series of a semigroup; provide

basic res ults concerning the structure of semigroups, and present elucidative

and useful examples, such as Rees matrix semigroups and Malcev nilpotent

semigroups. Fundamental notions from Semigroup Ring Theory are then

exposed: we deﬁne semigroup and contracted semigroup rings, followed by

some clarifying examples, and results concerning the structure of semisimple

semigroup algebras, with special attention given to semigroup algebras over

the rational ﬁeld, matrix semigroup rings and Munn algebras.

The reader who is well acquainted with all these concepts can concentrate

on the notation introduced.

Chapter 2 is part of a joint work ([33]) with Prof. Luciane Quoos,

from the Institute of Mathematics, UFRJ. The main result in this chapter is

an alternative procedure to compute the primitive central idempotents of a

group algebra of a ﬁnite nilpotent group over the complex ﬁeld, not relying

on the character table of the group, following methods previously applied

by Jespers and Leal to the rational case. As a partial result, we present

a formula for the primitive idempotents of the group algebra of the ﬁnite

abelian group over any ﬁeld, which permits to build all cyclic codes over a

given ﬁnite ﬁeld.

At ﬁrst, some notation is ﬁxed, and we state and prove a theorem that

yields a formula for the primitive idempotents in a semisimple group algebra

of a ﬁnite abelian group over an algebraically closed ﬁeld; though the result is

extremely useful and new, the proof is quite easy and short. Next, the same

result is extended to a semisimple group algebra of a ﬁnite abelian group

over an any ﬁeld, using the known method of Galois descent.

Then some technical deﬁnitions and facts, which appear in the literature

speciﬁc to the study of primitive central idempotents, are stated. A few

recent results on the subject are recalled, as they play an auxiliary role

to our main result. Finally, a fully internal description of the primitive

central idempotents of a group algebra of a ﬁnite nilpotent group over the

complex ﬁeld is presented. The class ical method for computing primitive

central idempotents in complex group algebras relies on the character table

of the group, whose construction has complexity growing exponentially with

the order of the group. Our tool depends only on a lattice of subgroups of the

given group, satisfying some intrinsic conditions. Though the complexity of

this new method is still unknown, it is a theoretic alternative to the classical

character method that might turn out to be simpler and faster to use.

The material in Cha pter 3 is based on a joint work ([9]) with Dr. Ann

Dooms, from the Department of Mathematics, Vrije Universiteit Brussel,

where I had the opportunity of spending one year of my Ph.D. under the

support of CNPq, Brasil. We deﬁne the new concept of the normalizer of a

semigroup in the unit group of its integral semigroup ring. Several known

results and properties on the normalizer of the trivial units in the unit group

of an integral group ring are shown to hold for the normalizer of a ﬁnite

inverse semigroup. This indicates that our concept of normalizer of a semi-

group be haves as desired, and might be suitable and helpful in investigations

on the isomorphism problem in semigroup rings and partial group rings, as

is analogously done with group rings. We also construct free groups in the

unit group of the integral semigroup ring of an inverse semigroup, using a

bicyclic unit and its image under an involution.

Introduction

We start by giving the deﬁnition of the normalizer of a semigroup in

the unit group of its integral semigroup ring. This deﬁnition coincides with

the normalizer of the trivial units in the case of an integral group ring and

behaves very much like it in inverse semigroups. These semigroups have a

natural involution, and semigroup rings of inverse semigroups are a wide

and interesting class containing, for instance, matrix rings and partial group

rings. This natural involution allows us to extend Krempa’s characterization

of the normalizer in group rings to a very useful property of the normalizer

of a semigroup. We will describe the torsion part of the normalizer and

investigate the double normalizer. Just like in group rings, the normalizer

of a semigroup contains the ﬁnite conjugacy center of the unit group of

the integral semigroup ring and the second center. It remains open what

the normalizer is in case the semigroup ring is not semisimple; we give an

example that provides some clues of how the normalizer might behave in

this case. This new concept of normalizer might be useful to tackle the

isomorphism problem for semigroup rings and partial group rings, and this

is an interesting path to follow in further studies.

The normalizer problem is then posed for integral semigroup rings, and

solved for ﬁnite Malcev nilpotent semigroups having a semisimple rational

semigroup ring. Like with integral group rings, we get that the normalizer

of a ﬁnite semigroup is a ﬁnite extension of the c enter of the semigroup ring.

Borel and Harish-Chandra showed the existence of free groups contained

in the unit group of an integral semigroup ring. As an additional consequence

of the investigation of the natural involution in a B randt semigroup, we

investigate the problem of constructing free groups in the unit group of an

integral semigroup ring using a bicyclic unit and its involuted image, following

Marciniak and Sehgal.

These are, so far, my contributions to the investigation of group rings.

Introduction

Chapter 1

Preliminaries

In this beginning chapter, we shall collect most of the needed background

in group and semigroup ring theory. For a more comprehensive approach, we

refer the reader to [25], [36], [4] and [30]. Those familiar with all the required

concepts can concentrate on the notation.

1.1 Group Ring Theory

1.1.1 Basic Deﬁnitions

Deﬁnition 1.1.1. Let G be a (multiplicative) group, and R be a ring with

identity 1. The group ring RG is the ring of all formal sums α =



g∈G

∈ R, with ﬁnite support supp(α) = {g ∈ G; α

= 0}. We say that α

the coeﬃcient of g in α. Two elements α and β in RG are equal if and only

if they have the s ame coeﬃc ients. The zero element in RG is the element

0 :=



g∈G

0g, and the identity in RG is the element 1 := 1e +



g∈G\{e}

(with e the identity in G, which shall be denoted henceforth by 1). The

sum α + β is the element



g∈G

(α

+ β

)g. The product αβ is the element



g∈G

g, where γ



x,y∈G

xy=g

, for each g ∈ G. If R is a commutative

ring, then RG is an R-algebra and is called a group algebra.

Deﬁnition 1.1.2. Let RG be a group ring. The augmentation mapping

of RG is the ring homomorphism ε : RG → R,



g →



. Its

kernel, denoted by ∆(G), is the augmentation ideal of RG. We have that

∆(G) =





g∈G

(g − 1); α

∈ R



If N is a normal subgroup of G, then there exists a natural homomorphism

: RG → R(G/N),



g →



gN. The kernel of this mapping,

denoted by ∆

(G, N), is the ideal of RG generated by {n − 1; n ∈ N}.

1.1. Group Ring Theory 2

We recall the deﬁnition of a ring with involution.

Deﬁnition 1.1.3. Let A be a commutative ring and R be an A-algebra. An

involution on R is an A-module automorphism τ : R → R such that, for all

x, y ∈ R

τ(xy) = τ (y)τ(x) and τ

(x) = x.

Let G be a group. If R is a ring with an involution τ, we can deﬁne an

involution ∗ on RG extending τ as

(



g∈G

g)∗ :=



g∈G

τ(α

−1

For instance, we can deﬁne an involution on CG, known as the classical

involution as (



g∈G

g)∗ :=



g∈G

−1

, where α

denotes the complex

conjugate of α

Now we need some concepts and results from Ring Theory that are rele-

vant in s tudying group rings.

Deﬁnition 1.1.4. Let R be a ring. The Jacobson radical of R, denoted by

Jac(R) or J (R), is deﬁned as the intersection of all maximal left ideals in R

if R = 0, or as (0) if R = 0.

Evidently, if R = 0, maximal ideals always exist by Zorn’s Lemma.

In the deﬁnition above, we used left ideals; we could similarly deﬁne the

right Jacobson radical of R. It turns out that both deﬁnitions coincide (see

[17, §4]), so the Jacobson radical of a ring is a two-sided ideal.

The Jacobson radical of a ring has many important prop e rties and shows

up in a number of theorems in Group and Semigroup Ring Theory. We shall

mention only the ones we need in this work.

Lemma 1.1.5. [25, Lemma 2.7.5] Let R be a ring. Then J (R/J (R)) = 0.

Deﬁnition 1.1.6. An element e in a ring R is said to be an idempotent

if e

= e. If e is an idempotent, e = 0 and e = 1, then e is a nontrivial

idempotent. Two distinct idempotents e and f in a ring R are said to be

orthogonal if ef = fe = 0. An idempotent e is said to be primitive if it cannot

be written as e = e



+ e



, with e



and e



nonzero orthogonal idempotents.

Deﬁnition 1.1.7. Let R be a ring, and M be a (left) R-module. M is called

a simple (or irreducible) (left) R-module if M is nonzero and M has no (left)

R-submodules other than (0) and M. A nonzero ring R is said to be a simple

ring if (0) and R are the only two-sided ideals of R.

Preliminaries

1.1. Group Ring Theory 3

Deﬁnition 1.1.8. Let R be a ring, and M be a (left) R-module. M is

called a semisimple (or completely reducible) (left) R-module if every (left) R-

submodule of M is a direct summand of M (i.e., for every (left) R-submodule

N of M, there is a (left) R-submodule N



such that M = N ⊕ N



Evidently, an analogous deﬁnition makes sense for rings (an equivalent

deﬁnition of semisimplicity for an Artinian ring R is Jac(R) = 0 [25, Theorem

2.7.16]) . It is worth observing that the notions of left and right semisimplicity

are equivalent for rings (see, for instance, [17, Corollary 3.7]).

If R is a semisimple ring, then the inner structure of R determines all the

simple R-modules, up to isomorphism.

Lemma 1.1.9. [25, Lemma 2.5.13] Let R be a semisimple ring, L be a

minimal left ideal of R and M be a simple R-module. We have that LM = 0

if and only if L  M as R-modules; in this case, LM = M.

Certain rings can be decomposed as a direct sum of ideals. Recall that

a ﬁeld F is a perfect ﬁeld if every algebraic extension of F is separable, or,

equivalently, char(F ) = 0 or char(F ) = p = 0 and F

= F .

Theorem 1.1.10 (Wedderburn–Malcev). [35, Theorem 2.5.37] Let R be

a ﬁnite dimensional algebra over a perfect ﬁeld F . Then

R = S(R) ⊕ J (R) (as a vector space over F ),

where J (R) is the Jacobson radical of R, and S(R) is a subalgebra of R

isomorphic to R/J (R).

In the Wedderburn–Malcev Theorem, notice that, since S(R)  R/J (R),

we have that S(R) is a semisimple algebra by Lemma 1.1.5.

The decomposition of a ring as a direct sums of ideals has a very important

particular case when the ring is semisimple.

Theorem 1.1.11. [25, Theorem 2.5.11] Let R =



i=1

be a decomposition

of a semisimple ring R as a direct sum of minimal two-sided ideals. Then

there exists a uniquely determined family {e

, . . . , e

} such that:

1. e

is a nonzero central idempotent, for i = 1, . . . , s;

2. e

= δ

, for i, j = 1, . . . , s, where δ

is Kronecker’s delta (in par-

ticular, e

and e

are orthogonal for i = j);

3. 1 =



i=1

;

Preliminaries

1.1. Group Ring Theory 4

4. e

cannot be written as e

= e



+ e



, where e



and e



are both nonzero

central orthogonal idempotents, for 1 ≤ i ≤ s.

Conversely, if there exists a family of idempotents {e

, . . . , e

} satisfying the

above conditions, then A

:= Re

are minimal two-sided ideals and R =



i=1

. The elements {e

, . . . , e

} are called the primitive central idempo-

tents of R.

Deﬁnition 1.1.12. The unique two-sided ideals of a semisimple ring R are

called the simple components of R

The following characterization of semisimple rings will be very important

for us:

Theorem 1.1.13 (Wedderburn–Artin). [17, Theorem 3.5] A ring R is

semisimple if and only if it is a direct sum of matrix algebras over divi-

sion rings, i.e., R  M

) ⊕ . . . ⊕ M

), where n

, . . . , n

are posi-

tive integers and D

, . . . , D

are division rings. The number s and the pairs

, D

), . . . , (n

, D

) are uniquely determined (up to permutations).

Now we can proceed to translate these ring theoretical concepts into valu-

able information about group rings.

Theorem 1.1.14 (Maschke). [25, Corollary 3.4.8] Let K be ﬁeld and G be

a ﬁnite group. The group algebra KG is semisimple if and only if char(K)

does not divide |G|.

The following special elements of a group ring play a major role both in

Group Ring Theory and in the present work in particular.

Deﬁnition 1.1.15. Let R be a ring with identity and let H be a ﬁnite

subset of a group G. Deﬁne the element



H of RG as



H :=



h∈H

h. If

H = a is a cyclic subgroup of G of ﬁnite order, we shall sometimes write a

instead of



a. If |H| is invertible in R, we may deﬁne the element



H of RG



H :=

|H|



h∈H

h. If {C

}

i∈I

is the set of conjugacy classes of G which

contain only a ﬁnite number of elements, then the elements



in RG are

called the class sums of G over R.

These constructions enable us to get, for instance, a basis for the center

of a group ring and idempotent elements.

Theorem 1.1.16. [25, Theorem 3.6.2] Let G be a group and R be a com-

mutative ring. The set {



}

i∈I

of all class sums of G over R is a basis of

Z(RG) over R.

Preliminaries

1.1. Group Ring Theory 5

Lemma 1.1.17. [25, Lemma 3.6.6] Let R be a ring with identity and let H

be a ﬁnite subgroup of a group G. If |H| is invertible in R, then



H is an

idempotent of RG. Moreover,



H is central if and only if H  G.

Group Representation Theory is a very powerful means to obtain new

results in Algebra. It is also a useful tool to “realize” an abstract group as

a concrete one. Some knowledge of Group Representation Theory is vital to

the understanding of group rings.

Deﬁnition 1.1.18. Let G be a group, R a commutative ring and V a free

R-module of ﬁnite rank. A representation of G over R, with representation

space V is a group homomorphism T : G → GL(V ), where GL(V ) denotes

the group of R-authomorphisms of V . The rank of V is called the degree

of T and denoted by deg(T ). Fixing an R-basis of V , we can deﬁne an

isomorphism between GL (V ) and GL

(R), with n = deg(T ), where GL

(R)

denotes the group of n × n invertible matrices with coeﬃcients in R. We can

thus consider the induced group homomorphism T : G → GL

(R), in which

case we talk about a matrix representation.

Two representations T : G → GL(V ) and T



: G → GL(W ) of G over R

are equivalent representations if there exists an isomorphism of R-modules

φ : V → W such that T



(g) = φ ◦ T (g) ◦ φ

−1

, for all g ∈ G. A representation

T : G → GL (V ) is an irreducible representation if V = 0 and if V and 0 are

the only invariant subspaces of V under T .

There is a strong connection between group representations and modules,

in which group rings play a major role, as it may be seen in the following

proposition:

Proposition 1.1.19. [25, Propositions 4.2.1 and 4.2.2] Let G be a group

and R be a commutative ring with identity. There is a bijection between the

representations of G over R and the (left) RG-modules which are free of ﬁnite

rank over R:

• given a representation T : G → GL(V ) of G over R, associate to

it the (left) RG-module M constructed from V by keeping the same

additive structure and deﬁning the product of v ∈ V by α ∈ RG as

αv :=



g∈G

T (g)(v);

• if M is a (left) R G-module which is free of ﬁnite rank over R, deﬁne

the representation T : G → GL(M) ; T (g) : m → gm.

Two representations of G over R are equivalent if and only if the corre-

sponding (left) RG-modules are isomorphic. Also, a representation is irre-

ducible if and only if the corresponding (left) RG-module is simple.

Preliminaries

1.1. Group Ring Theory 6

The case when a group is represented over a ﬁeld is of particular inter-

est. Historically, this was the ﬁrst case to be studied and, therefore, most

applications were developed in this context.

The notion of character is of fundamental importance in Group Represen-

tation Theory and in Group Theory. It also shows up in the study of group

rings; in particular, there is a well-known formula for computing idempo-

tents of complex group algebras of ﬁnite groups that completely relies on

characters.

Deﬁnition 1.1.20. Let T : G → GL(V ) be a representation of a group G

over a ﬁeld K, with representation space V . The character χ of G aﬀorded

by T is the map χ : G → K; g → tr(T (g)), where tr(T (g)) is the trace of the

matrix associated to T (g) with respect to any basis of V over K. If T is an

irreducible representation, then χ is called an irreducible character

Proposition 1.1.21. [25, Proposition 5.1.3] Let G be a ﬁnite group and K be

a ﬁeld such that char(K)  |G|. Consider χ

, . . . , χ

the characters aﬀorded by

a complete set T

, . . . , T

of inequivalent irreducible representations of G over

K. Then the set of all characters of G over K is the set {χ =



i=1

; n

∈

Z, i = 1, . . . , r}.

Next, we give an example of a very special representation and the corre-

sponding character, which will be useful in the sequel.

Example 1.1.22 (Regular Representation and Regular Character).

Let G be a ﬁnite group of order n. Consider the representation T : G →

GL(CG) that associates to each g ∈ G the linear map T

: x → gx. This

is called the regular representation of G over C. Denote by ρ the character

aﬀorded by T , i.e., the regular character.

Regard T as the matrix representation obtained by taking the e lements

of G in some order as an R-basis of RG. It is clear that the image of each

g ∈ G is a pe rmutation matrix. Notice that if g = 1, then gx = x, so all the

elements on the diagonal of matrix T (g) are equal to zero. Hence,

ρ(g) =



0 , if g = 1

|G|, , if g = 1.

We know that

CG  M

⊕ . . . ⊕ (CG)e

(Theorem 1.1.13 and Theorem 1.1.11), with {e

, . . . , e

} the primitive central

idempotents of CG, and that, for all i = 1, . . . , s, M



Preliminaries

1.1. Group Ring Theory 7

⊕ . . . ⊕ L

, where L

denotes the irreducible (left) CG-module consisting

of n

× n

matrices having complex elements on the j

column and zeros

elsewhere. Clearly, by Lemma 1.1.9, for all j = 1, . . . , n

, L

 L

(as CG-

modules), which has dimension n

over C. So, CG  n

⊕ . . . ⊕ n

If T

denotes the irreducible representation of G over C corresponding to

and χ

denotes the character T

aﬀords, then we have that T =



i=1

and ρ =



i=1

. Since n

= deg(T

) = χ

(1) (because χ

(1) = tr (T

(1)) =

tr(I

), where I

is the n

× n

identity matrix), it follows that

ρ =



i=1

(1)χ

(*).

In the isomorphism CG  M

(C), each idempotent

corresponds to the element (0, . . . , 0, I

, 0, . . . , 0), with I

the identity

matrix in M

(C). It is clear that T

) is the linear function deﬁned in L

by multiplication by the identity element, i.e., it is the identity function on

the simple component M

(C). Since e

= δ

, it follows that

) =



0 , if i = j,

, if i = j;

and

) =



0 , if i = j,

tr(I

) = deg(T

) , if i = j.

Lemma 1.1.23. Let G be a group and χ be a character aﬀorded by a rep-

resentation of G. Then χ is constant on each of the conjugacy classes of

Thus, the following deﬁnition makes sense:

Deﬁnition 1.1.24. Let G be a group and C

, . . . , C

be its conjugacy classes.

Choose, for each i, an arbitrary x

∈ C

. The matrix (χ

)) is called the

character table of G.

The case of group characters over the complex ﬁeld will be extremely

important for us. We would like to know the dimensions of the complex

character table of a group. Notice that we learn, from example 1.1.22, that

the number of irreducible characters of a group G is equal to s, the number

of simple components in the Wedderburn–Artin decomposition of CG. In

fact, more is known:

Preliminaries

1.1. Group Ring Theory 8

Proposition 1.1.25. [25, Proposition 3.6.3, Theorem 4.2.7] Let G be a ﬁ-

nite group and K be an algebraically closed ﬁeld such that char(K)  |G|.

Then the number of simple components of KG is equal to the number of

conjugacy classes of G, which equals the number of irreducible nonequivalent

representations of G over K.

Remark 1.1.26. Actually, the above proposition is still true in a slightly

more general setting: whe n K is a splitting ﬁeld for G (see [25, Deﬁnition

3.6.4]).

Let A be a ring (with identity). We recall that we denote by U(A) the

multiplicative group of units of A, i.e.,

U(A) := {x ∈ A; xy = yx = 1, for some y ∈ A}.

There are not many known methods for constructing units in group rings,

most of them being either elementary or very old. Therefore, describing units

in group rings is a very active and important ﬁeld of research.

Deﬁnition 1.1.27. Let R be a ring with identity and G be a group. An

element in the group ring RG of the form rg, where r ∈ U(R) and g ∈ G, is

called a trivial unit of RG, its inverse being the element r

−1

For instance, the trivial units in ZG are the ele ments of ±G. Generally

speaking, group rings do have nontrivial units, though these may be hard to

ﬁnd.

Let A be a ring with zero divisors. Take x, y ∈ A \ {0} such that xy = 0.

Then, for an arbitrary t ∈ A, η := ytx is a square zero element. Thus, 1 + η

is a unit, with inverse 1 − η. A very special and important case occurs when

A is an integral group ring.

Deﬁnition 1.1.28. Let G be a group. Consider a, b ∈ G, with a of ﬁnite

order, and deﬁne

a,b

:= 1 + (1 − a)ba.

Since (1 − a)a = 0, u

a,b

is a unit in ZG, called a bicyclic unit.

Notice that the deﬁnition of bicyclic units still makes sense for integral

semigroup rings (see Section 1.2).

Proposition 1.1.29. [25, Proposition 8.1.6] Let g, h be elements of a group

G, with o(g) < ∞. Then, the bicyclic unit u

g,h

is trivial if and only if h

normalizes g and, in this case u

g,h

= 1.

Preliminaries

1.1. Group Ring Theory 9

Next, we deﬁne a type of unit that is related to the classical involution

on an integral group ring.

Deﬁnition 1.1.30. Let G be a group. An element u ∈ ZG is said to be a

unitary unit if uu

∗

= u

∗

u = 1, where ∗ is the c lassical involution on ZG (see

Deﬁnition 1.1.3).

Now we recall some deﬁnitions from Group Theory that will be helpful

when studying the unit group of group rings and semigroup rings (see [34]).

Deﬁnition 1.1.31. Let G be a group. Consider the upper central series of

{1} = Z

(G) ≤ Z

(G) ≤ . . . ,

deﬁned inductively as Z

(G) := Z(G) and Z

(G), the n

center of G, is the

only subgroup of G such that

(G)/Z

n−1

(G) = Z(G/Z

n−1

(G)).

Notice that u ∈ Z

n+1

(G) if and only if (u, g) ∈ Z

(G), for all g ∈ G, where

(u, g) := u

−1

ug is the commutator of u and g.

The union

∞

(G) :=



(G)

is called the hypercenter of G. If there exists m ∈ N such that Z

∞

(G) =

(G) and m is the smallest possible number with this property, then m is

called the central height of G.

Let N and H be subsets of G. The centralizer of H in N is deﬁned by

(H) := {g ∈ N; gh = hg, ∀h ∈ H}.

The ﬁnite conjugacy center of G (or FC center of G) Φ(G) is the set of

all elements of G that have a ﬁnite number of conjugates in G, i.e.,

Φ(G) := {g ∈ G; |C

| < ∞} = {g ∈ G; (G : C

(g)) < ∞},

where C

denotes the G-conjugacy class of g ∈ G. It is well known ([25,

Lemma 1.6.3]) that Φ(G) is a characteristic subgroup of G .

Let H be a subgroup of G and g, h ∈ G. We denote by H

the conjugate

of H by g, i.e .,

= gHg

−1

and we denote by h

the conjugate of h by g, i.e.,

= ghg

−1

Preliminaries

1.1. Group Ring Theory 10

1.1.2 Some Results

In this subsection, we recall some results that will be helpful in the sequel.

Although these are basic facts from Group Ring Theory, we include their

proofs for the sake of completeness.

The next theorem deals with the problem of ﬁnding the Wedderburn–

Artin decomposition (according to Theorem 1.1.13) of a semisimple group

algebra KG of a ﬁnite abelian group G.

Theorem 1.1.32. [32, Perlis–Walker] Let G be a ﬁnite abelian group of

order n, and K be a ﬁeld such that char(K)  n. Then

KG 



d|n

K(ζ

where a

K(ζ

) denotes the direct sum of a

copies of K(ζ

), ζ

are primitive

roots of unity of order d and a

= n

/[K(ζ

) : K], with n

denoting the

number of elements of order d in G.

Proof. Let us ﬁrst analyse the case w hen G = a; a

= 1 is a cyclic group of

order n.

Consider the map φ : K[X] → KG; f → f(a). We have that φ is a ring

epimorphism with kernel (X

− 1); thus KG 

K[X]

−1)

. Consider X

− 1 =



d|n

, the decomposition of X

−1 in cyclotomic polynomialnomials Φ

K[X], i.e., Φ



(X−ζ

), where ζ

runs over all the primitive roots of unity

of order d, for all d|n. For each d, let Φ



i=1

be the decomposition of

in irreducible polynomials in K[X]. So,

− 1 =



d|n



i=1

Since char(K)  n, it follows that X

− 1 is a separable polynomial, i.e.,

all the f

’s are distinct irreducible polynomials. By the Chinese Remainder

Theorem, it follows that

KG 



d|n



i=1

K[X]

)

Now, for each d

, we have that

K[X]

)

 K(ζ

), with ζ

a root of f

. Thus,

KG 



d|n



i=1

K(ζ

Preliminaries

1.1. Group Ring Theory 11

and ζ

are primitive roots of unity of orders d dividing n. So, for a ﬁxed d,

we have that K(ζ

) = K(ζ

), for any i, j = 1, . . . , a

, and we may write

KG 



d|n

K(ζ

with ζ

a primitive root of unity of order d. Also, deg(f

) = [K(ζ

) : K],

for all i = 1, . . . , a

, so, taking degrees in the decomposition of Φ

, we have

that φ(d) = a

[K(ζ

) : K], where φ denotes Euler’s totient function. It

is well known that, since G is a cyclic group of order n, for any divisor d

of n, the number n

of elements of order d in G is precisely φ(d). Hence,

= n

/[K(ζ

) : K].

Suppose now that G is a ﬁnite abelian noncyclic group. We proceed by

induction on the order of G. So assume the result holds for any abelian group

of order less than n.

Using the Structure Theorem of Finite Abelian Groups, we can write

G = G

× H, with H a cyclic group of order n

and |G

| = n

< n.

By the induction hypothesis, we can write KG





K(ζ

), with

[K(ζ

):K]

and n

denoting the number of elements of order d

in G

Therefore,

KG  K(G

× H)  (KG

)H  (



K(ζ

))H 



(K(ζ

)H)

(because, for any ring R and any groups G and H, it holds that R(G ×

H)  (RG)H, and, if R = ⊕

i∈I

, with {R

}

i∈I

a family of rings, then

RG  ⊕

i∈I

G). Decomposing each direct summand, we get

KG 



K(ζ

, ζ

with a

= n

/[K(ζ

, ζ

) : K(ζ

)] and n

denoting the number of elements

of order d

in H. Taking d := lcm(d

, d

), it follows that K(ζ

) = K(ζ

, ζ

Set a



lcm(d

)=d

and let us see that the result follows. In fact,

since [K(ζ

) : K] = [K(ζ

, ζ

) : K(ζ

)][K(ζ

) : K], we have that

[K(ζ

) : K] =



lcm(d

)=d

[K(ζ

, ζ

) : K(ζ

)][K(ζ

) : K] =



lcm(d

)=d

From G = G

× H, each element in g ∈ G may be written as g = g

with g

∈ G

and h ∈ H and, since G is abelian, o(g) = lcm(o(g

), o(h)).

Hence,



lcm(d

)=d

= n

, the number of elements of order d in G,

and a

= n

/[K(ζ

) : K]. Therefore KG 



d|n

K(ζ

)

Preliminaries

1.1. Group Ring Theory 12

If one knows the Wedderburn–Artin decomposition of a semisimple ring,

it is only natural to search for its primitive central idempotents (Theorem

1.1.11). In Chapter 2, we develop methods to ﬁnd primitive central idempo-

tents of semisimple group algebras of ﬁnite abelian groups and of complex

group algebras of ﬁnite nilpotent groups. In the latter case, there already

exists a classical method to compute the primitive central idempotents that

relies on the group’s character table.

The character table of a group provides the primitive central idempotents

of its complex group algebra by means of the following formula.

Theorem 1.1.33. [25, Theorem 5.1.11] Let G be a ﬁnite group, and

, . . . , χ

be all the irreducible complex characters of G. For i = 1, . . . , r,

deﬁne

(1)

|G|



g∈G

−1

)g.

Then e

, . . . , e

are the primitive central idempotents of the complex group

algebra CG.

Proof. For each i = 1, . . . , r, we may write e



g∈G

g. Evaluating the

regular character ρ on e

, we get ρ(e

) =



g∈G

ρ(g) = α

|G|. Thus, for

any x ∈ G, we have that ρ(x

−1

) =



g∈G

ρ(g) = α

|G|. From Example

1.1.22, (∗) it follows that α

|G| = ρ(x

−1

) =



j=1

(1)χ

−1

). Consider

the representation associated to the character χ

. We get that

−1

) = T

−1

) = T

−1

) = T

−1

) = 0,

for i = j, and thus

−1

) = χ

−1

) = 0,

for i = j. As a result,

|G|

(1)χ

−1

for all x ∈ G. Hence, from e



g∈G

g, the desired formula follows.

Now we proceed to state some results about trivial units (see Deﬁnition

1.1.27) in integral group rings that will be very relevant in the study of the

normalizer of integral semigroup rings in Chapter 3.

The next result states that the trivial units are the only unitary units

(see Deﬁnition 1.1.30); it has an elementary proof and, nevertheless, is very

useful in the study of units in group rings.

Preliminaries

1.1. Group Ring Theory 13

Proposition 1.1.34. Let G be a group and γ =



g∈G

g ∈ ZG. We have

that γγ

∗

= 1 if and only if γ ∈ ±G.

Proof. If γ = ±g, for g ∈ G, then γ

∗

= ±g

−1

= γ

−1

Conversely, if γγ

∗

= 1, then we have that

1 =



g∈G

1 +



g,h∈G

g=h

−1

Thus, 1 =



g∈G

1 and 0 =



g,h∈G

g=h

−1

. Since γ

∈ Z for all g ∈ G,

this implies that γ

= ±1 for a unique g

∈ G and γ

= 0 for all g = g

Hence, γ = ±g

The next result, due to Berman and Higman, is also valid for inﬁnite

groups; however, we shall only need it for ﬁnite groups, and, in this case,

there is an independent proof.

Lemma 1.1.35 (Berman–Higman). Let γ =



g∈G

g be a unit of ﬁnite

order in the integral group ring ZG of a ﬁnite group G. If γ

= 0 then

γ = γ

= ±1.

Proof. Let n = |G| and suppose γ

= 1, for some positive integer m. Con-

sider the regular representation T and the regular character ρ (see Example

1.1.22) on the group algebra CG, and regard ZG as a subring of CG. We

have that

ρ(γ) =



g∈G

ρ(g) = γ

Since γ

= 1, we have that T (γ)

= T (γ

) = T (1) = I; thus T (γ) is a

root of the p olynomial X

− 1 = 0, which is separable. So there is a basis

of CG such that T (γ) is an n × n diagonal matrix, with m

roots of unity

in the diagonal and zeros elsewhere.

Hence,

ρ(γ) = tr(T (γ)) =



i=1

= nγ

and, taking absolute values,

n|γ

| ≤



i=1

|ζ

| = n.

Now, n|γ

| ≥ n, for γ

∈ Z, then we must have |γ

| = 1 and also



i=1

|ζ

| =



i=1

|, which happens if and only if ζ

= . . . = ζ

Therefore, nγ

= nζ

, and consequently γ

= ζ

= ±1. So, T (γ) = ±I

and γ = ±1.

Preliminaries

1.2. Semigroup Ring Theory 14

The following corollaries of the Berman–Higman Lemma will be extremely

helpful for us.

Corollary 1.1.36. Let A be a ﬁnite abelian group. Then the group of torsion

units of the integral group ring ZA is the group of trivial units ±A.

Proof. Let γ =



g∈A

g ∈ ZA be a unit of ﬁnite order. Suppose that

= 0, for some g

∈ A. Then, due to the commutativity of A, γg

−1

is also

a unit of ﬁnite order in ZA and (γg

−1

)

= γ

= 0. By Lemma 1.1.35, we

have that γg

−1

= ±1, i.e., γ = ±g

Corollary 1.1.37. Let G be a ﬁnite group. Then the group of all torsion

central units of the integral group ring ZG is the group of the central trivial

units ±Z(G).

1.2 Semigroup Ring Theory

Before getting to the topic of semigroup rings itself, we will need some

theory about semigroups.

1.2.1 Prerequisite: Semigroup Theory

Some notions on Semigroup Theory are generalizations of concepts from

Group Theory; however, most of them resemble Ring Theory.

Deﬁnition 1.2.1. A semigroup is a nonempty set with an associative bi-

nary operation, which will be denoted multiplicatively by juxtaposition of

elements.

An element θ of a semigroup S is called a zero element if sθ = θs = θ , for

all s ∈ S. A null semigroup is a semigroup with zero in which the product

of any two elements is the zero element.

An element 1 of a semigroup S is called an identity if s1 = 1s = s, for all

s ∈ S. A monoid is a semigroup that has an identity.

An element s in a monoid S is said to be an invertible element (or a unit)

if there exists s



∈ S such that ss



= s



s = 1; the element s



is called the

inverse element of s and is denoted s

−1

. The unit group of a monoid S is the

group U(S) := {s ∈ S; sr = rs = 1 , for some r ∈ S}. A grou p S is a monoid

in which every element is invertible, i.e., S = U(S). A group S is said to be

an abelian group (or a commutative group) if sr = rs, for all s, r ∈ S.

A semigroup homomorphism is a function f : S −→ T from a semigroup

S to a semigroup T such that f (rs) = f(r)f(s), for all r, s ∈ S.

Preliminaries

1.2. Semigroup Ring Theory 15

If S is a semigroup, then we denote by S

the smallest monoid containing

S. So



S , if S already has an identity element;

S ∪ {1} , if S does not have an identity element,

with s1 = 1s = s, for all s ∈ S

. Similarly, we denote by S

the smallest

semigroup with a zero containing S. So



S , if S already has a zero element;

S ∪ {θ} , if S does not have a zero element,

with sθ = θs = θ, for all s ∈ S

. In particular, for a group G, we say that

is a group with zero.

Adjoining a zero normally simpliﬁes arguments, while adjoining an iden-

tity is often useless, as the structure theory of semigroups relies on s ubsemi-

groups and ideals, which do not have an identity in general.

The following example describes a kind of matrix semigroup that is of

utmost importance in the algebraic theory of semigroups.

Example 1.2.2 (Rees Matrix Semigroup). Let G be a group, G

G ∪ {θ} be the group with zero obtained from G by the adjunction of a zero

element θ (as in Deﬁnition 1.2.1), and I and M be arbitrary nonempty sets.

By an I × M matrix over G

we mean a mapping A : I × M → G

, for which

we use the notation a

i,m

:= A(i, m), for (i, m) ∈ I × M, and (a

i,m

) := A.

By a Rees I × M matrix over G

we mean an I × M matrix over G

having at most one nonzero element. For g ∈ G, write (g)

i,m

, with i ∈ I

and m ∈ M, for the I × M matrix having g in the (i, m)-entry, its remaining

entries being θ. For any i ∈ I and m ∈ M, the expression (θ)

i,m

denotes the

I × M zero matrix, which will be also be denoted by θ.

Now let P be a ﬁxed arbitrary M × I matrix over G

. We deﬁne a

multiplication operation in the set of all Rees I × M matrices over G

AB := A ◦ P ◦ B, where ◦ denotes the usual multiplication of matrices and,

in p erforming this, we agree that, for g ∈ G

, θ + g = g = g + θ. We call P

the sandwich matrix with respect to this multiplication. Clearly, the set of

all Rees I × M matrices over G

is closed under this operation, which is also

associative. So, we can consider the Rees I × M matrix semigroup over the

group with zero G

with sandwich matrix P and denote it by M

(G, I, M, P ).

When the sets I and M are ﬁnite, say |I| = n and |M| = m, we will write

(G, I, M, P ) simply as M

(G, n, m, P ).

In fact, this type of semigroups is very natural, for instance:

Preliminaries

1.2. Semigroup Ring Theory 16

1. In the ring of integral n × n matrices M

(Z), denote by e

i,j

the n × n-

matrix with 1 as the (i, j)-entry and zeros elsewhere. We call e

i,j

matrix unit . We may multiply matrix units in the following way:

i,j

k,l



i,l

, if j = k,

0, if j = k.

The matrix units in M

(Z) and the n × n zero matrix form, with

this multiplication, the matrix semigroup M

({1}, n, n, I

), where I

denotes the n × n-identity matrix.

2. Let G be a group with identity 1 and P =





. The matrix semi-

group M

(G, 1, 2, P ) is isomorphic to the semigroup G

∪ G

∪ {θ},

with G

and G

isomorphic copies of G such that G

⊆ G

and

⊆ G

Now we deﬁne some special subsets of semigroups having a certain alge-

braic structure.

Deﬁnition 1.2.3. A subsemigroup of a semigroup is a nonempty subset

which is closed under multiplication. A submonoid of a semigroup is a sub-

semigroup with an identity. A subgroup of a semigroup is a subsemigroup

that is a group.

If T is a nonempty subset of a semigroup S, we write T  for the sub-

semigroup generated by T (if T is ﬁnite, say T = {t

, . . . , t

}, we often write

t

, . . . , t

 instead of T ). A semigroup S is said to be a cyclic semigroup

if S = x for some x ∈ S. An element x of a semigroup S is a periodic

element if x is ﬁnite. A semigroup is a periodic semigroup if every cyclic

subsemigroup is ﬁnite.

Note that the identity of a subgroup G of a semigroup S need not to be

the identity of S (actually, S may not have an identity at all). As as example,

consider S := M

(G, n, n, I

), where I

denotes the n × n identity matrix

and G is any group; S has no identity element. But M := {s = ge

1,1

∈

(G, n, n, I

); g ∈ G} is a subgroup of S with identity 1e

1,1

Deﬁnition 1.2.4. An element e in a semigroup S is called an idempotent

if e = e

. We write E(S) for the set of idempotent elements of a semigroup

S. The set E(S) has a natural partial order: e ≤ f ⇐⇒ ef = fe = e.

An idempotent e in a semigroup S is said to be primitive if it is a nonzero

idempotent and if it is minimal with respect to the partial order in E(S).

Preliminaries

1.2. Semigroup Ring Theory 17

Notice that the notion of primitive idempotent in a ring (Deﬁnition 1.1.6)

is equivalent to the one given above.

Observe that a ﬁnite cyclic semigroup always contains an idempotent.

In fact, let s be a ﬁnite cyclic semigroup. So there exist positive integers

n and k so that s

n+k

= s

. Hence, s

n+vk

= s

, for any positive integer v.

In particular, s

n(1+k)

= s

. So the semigroup s contains an element a so

that a

= a, for some integer m ≥ 2. If m = 2, then a is an idempotent.

Otherwise, a

m−1

, a

m−2

∈ s and

m−1

)

= a

m−1

= (a

m−1

a)(a

m−2

) = a

m−2

= aa

m−2

= a

m−1

and then a

m−1

is an idempotent.

Deﬁnition 1.2.5. Let I be a nonempty subset of a semigroup S. We say I

is a right ideal of S if xs ∈ I, for all s ∈ S and x ∈ I, i.e., IS

⊆ I. A left

ideal is deﬁned analogously. We call I an ideal of S if it is a left and a right

ideal of S.

For a ∈ S, the ideal generated by a is deﬁned as J

:= S

= SaS ∪

Sa ∪ aS ∪ {a}.

Deﬁnition 1.2.6. A semigroup S is said to be a regular semigroup if it

satisﬁes the Von Neumann regularity condition, i.e., for every s ∈ S, there

exists x ∈ S such that sxs = s. A semigroup is said to be an inverse

semigroup if it is regular and its idempotents commute; or, equivalently,

every principal right ideal and every principal left ideal of S has a unique

idempotent generator; or, also equivalently, if for every s ∈ S, there exists a

unique x ∈ S such that sx s = s and xsx = x ([4, Theorem 1.17]).

Let S be a semigroup. Deﬁne the the center of S as the subset Z(S) =

{x ∈ S; xs = sx, ∀s ∈ S}.

If a semigroup S has a minimal ideal K, then K is called a kernel of S.

Clearly, any two distinct minimal ideals of S are disjoint. Since two ideals A

and B of S always contain their set product AB, it follows that S can have

at most one kernel K. Notice that S may not have a kernel at all (this is the

case, for instance, if S is an inﬁnite cyclic semigroup). If S has a kernel K,

K may be characterized as the intersection of all ideals of S, because K is

contained in every ideal of S.

Let e be an idempotent in a semigroup S. Then eSe = {ese; s ∈ S} is

a submonoid of S with identity element e. Now, eSe coincides with {x ∈

S; ex = xe = x}, the set of elements of S for which e is an identity. Consider

:= U(eSe), the unit group of the monoid eSe. Then H

is a subgroup

of S and it is the largest subgroup of S for which e is the identity. Such

subgroups are called the maximal subgroups of S. Notice that all the maximal

Preliminaries

1.2. Semigroup Ring Theory 18

subgroups of S are isomorphic. There is a one-to-one correspondence between

the idempotents e and maximal subgroups H

of a semigroup S, since e is

the unique idempotent element of H

. H

contains every subgroup of S that

meets H

. Thus, distinct maximal subgroups are disjoint.

Now, the Rees matrix semigroups deﬁned in Example 1.2.2 are important

in characterizing some special kinds of semigroups.

Lemma 1.2.7. [4, Lemma 3.1] Let G be a group, I and M be arbitrary

nonempty sets, and P be a M × I matrix over G

. Then the Rees I ×

M matrix semigroup M

(G, I, M, P ) over G

with sandwich matrix P is a

regular semigroup if and only if each row and each column of P contains a

nonzero entry. In such a case, P is said to be a regular matrix.

Proposition 1.2.8. [30, Lemma 1.4] Let G be a group, I and M be arbitrary

nonempty sets, and P be a M × I matrix over G

. Then the nonzero idempo-

tents of the Rees I ×M matrix semigroup M

(G, I, M, P ) over G

with sand-

wich matrix P are precisely the elements e = (p

−1

j,i

)

i,j

, with p

j,i

= θ. Deﬁne

i,j

:= {(g)

i,j

; g ∈ G

, p

j,i

= θ}. Then all the S

i,j

\ {θ} together with {θ} are

the maximal subgroups of M

(G, I, M, P ). In fact, eM

(G, I, M, P )e  S

i,j

Notice that, on the proposition above, all the S

i,j

are isomorphic to G

Deﬁnition 1.2.9. Let S be a semigroup. An equivalence relation ρ on S is

called a right congruence relation if, for all a, b, c ∈ S, it holds that aρb implies

acρbc. An equivalence relation ρ on S is called a left congruence relation if,

for all a, b, c ∈ S, it holds that aρb implies caρcb. An equivalence relation

ρ on S is a congruence relation if it is both a right and a left congruence

relation. We denote by S/ρ the set of the equivalence classes and by a the

equivalence class containing the element a ∈ S. S/ρ becomes a semigroup

with multiplication deﬁned as ab = ab. We call S/ρ the factor semigroup of

S modulo ρ.

Let S be a semigroup and I be an ideal in S. Deﬁne ρ the Rees congruence

modulo I on S as aρb ⇐⇒ a = b or a, b ∈ I. We write S/I for S/ρ and call

this the Rees factor semigroup of S modulo I. As a convention, the Rees

factor S/∅ is deﬁned to be S (even though the empty set ∅ is not an ideal).

Clearly, for a semigroup S and a congruence relation ρ, there is a natural

semigroup homomorphism S → S/ρ; a → a. Congruence relations on groups

yield the concept of normal subgroups.

Let S be a semigroup and I be an ideal in S. Notice that the equivalence

classes of S/I are I itself and every one-element set {a}, with a ∈ S \ I.

Thus, as a set, S/I may be identiﬁed with S \ I with an element θ adjoined,

Preliminaries

1.2. Semigroup Ring Theory 19

and such that aθ = θa = θ, so θ is actually a zero element. Intuitively,

when we pass from S to S/I, we identify all the elements of I with θ and

the nonzero elements of S/I correspond with S \ I; therefore, we will usually

denote the nonzero elements of S/I as a (with a ∈ S \ I) instead of a. There

is a one-to-one correspondence between the ideals of S containing I and the

ideals of S/I.

In analogy with what happens in groups and rings, we have the following

lemma:

Lemma 1.2.10. Let S be a semigroup and I, J be ideals of S such that

I ⊆ J. Then J/I is an ideal of S/I and (S/I)/(J/I)  S/J.

We introduce, in analogy to simple rings, special kinds of semigroups that

are “indecomposable” in some sense.

Deﬁnition 1.2.11. A semigroup is called a simple semigroup if it has no

ideals other than itself.

A semigroup S with a zero θ is said to be a 0-simple semigroup if S has

no ideals other than S and {θ}, and S

= S, or, equivalently S has no proper

ideals other than {θ}, and S is not a null semigroup of cardinality two.

A semigroup is called a completely 0-simple semigroup if it is 0-simple

and it contains a primitive idempotent (see Deﬁnition 1.2.4).

A semigroup is said to be a Brandt semigroup if it is a completely 0-simple

inverse semigroup.

Notice that if S is a semigroup having zero element θ, then {θ} is always

an ideal of S. Thus, the deﬁnition of simple semigroup is not very interesting.

Completely 0-simple semigroups, on the other hand, will turn out to be the

building blocks of the structure theory of semigroups.

Let us characterize 0-simple semigroups and completely 0-simple semi-

groups in more detail:

Lemma 1.2.12. [4, Lemma 2.28] Let S be a semigroup with zero θ, and

such that S = θ. Then S is 0-simple if and only if SaS = S for every a ∈ S,

a = θ.

Lemma 1.2.13. ([4, §2.7, Exercise 11] A semigroup S with zero is com-

pletely 0-simple if and only if all of the following conditions are satisﬁed:

1. S is regular;

2. every nonzero idempotent of S is primitive;

3. if e and f are nonzero idempotents of S, then eSf = θ.

Preliminaries

1.2. Semigroup Ring Theory 20

The next theorem characterizes completely 0-simple semigroups and also

Brandt semigroups.

Theorem 1.2.14. [4, Theorem 3.5, Theorem 3.9] A semigroup is completely

0-simple if and only if it is isomorphic to a regular Rees matrix semigroup

over a group with zero.

Furthermore, a semigroup is a Brandt semigroup if and only if it is iso-

morphic to a Rees matrix semigroup M

(G, M, M, I

|M|

) over a group with

zero G

with the M × M identity matrix I

|M|

as sandwich matrix.

Deﬁnition 1.2.15. Let S be a semigroup with zero. Deﬁne the equivalence

relation J in S by xJ y ⇐⇒ J

= J

(see Deﬁnition 1.2.5), i.e., x and y

generate the same ideal in S. We use the notation J(x) for the J -class of

S containing x, i.e., J(x) is the set of elements that generate the ideal J

Two elements are said to be J -equivalent if they determine the same J -

class. Let I

denote the set of elements of J

that do not generate J

, i.e.,

:= {y ∈ J

; J

 J

}. So I

is an ideal of S and J

\ I

= J(x). The

quotient S

:= J

is called a principal factor of S

By a principal series of S we mean a strictly decreasing chain of ideals S

of S, begining with S and ending with {θ}, and such that there is no ideal of

S strictly between S

and S

i+1

, for i = 1, . . . , m. The factors of a principal

series of S are the Rees factors S

i+1

Remark 1.2.16. Notice that the deﬁnition of principal series with which we

are working is slightly diﬀerent from that of [4]. Our series ends with the

zero ideal rather than the empty set; so we will always adjoin a zero to the

semigroup before considering its principal series.

From the following lemma, it becomes clear why 0-simple and completely

0-simple semigroups are said to be “building blocks” for all semigroups.

Lemma 1.2.17. [4, Lemma 2.39] Each principal factor of any semigroup

(with zero) is either 0-simple or null.

Theorem 1.2.18. [4, Theorem 2.40] Let S be a semigroup (with zero), hav-

ing principal series S = S

⊃ S

⊃ · · · ⊃ S

⊃ S

m+1

= {θ}. Then its factors

are isomorphic in some order to the principal factors of S. In particular, any

two principal series of S have isomorphic factors.

Corollary 1.2.19. [4, Corollary 2.56] Any periodic (in particular, any ﬁnite)

0-simple semigroup is completely 0-simple.

Theorem 1.2.20. Every ﬁnite semigroup with zero has a principal series

and the principal factors are either completely 0-simple or null semigroups.

Preliminaries

1.2. Semigroup Ring Theory 21

To end this section, we give an example of a well known class of semi-

groups.

Example 1.2.21 (Malcev Nilpotent Semigroups). [21, 26, 30] Let S

be a semigroup. Consider, for x, y in S, and w

, w

, . . . , w

, . . . ∈ S

, the

following sequence deﬁned inductively:

= x,

= y,

and for n ≥ 0

n+1

= x

n+1

= y

n+1

If x

= y

, for all x

, y

∈ S and all w

in S

, and n is the least positive

integer with this property, then S is said to be a Malcev nilpotent semigroup

of class n.

Finite Malcev nilpotent semigroups have been classiﬁed by Okni´nski in

[31]. It is interesting to observe that a group is Malcev nilpotent of class n if

and only if it is nilpotent of class n in the classical sense [30, Theorem 7.2].

We have that a Brandt semigroup which is a matrix semigroup over a

nilpotent group (Theorem 1.2.14) is Malcev nilpotent. Actually, these are

the only Malcev nilpotent completely 0-simple semigroups [16, Lemma 2.1].

Hence, the completely 0-simple principal factors of a ﬁnite Malcev nilpotent

semigroup are Brandt semigroups, which are of the form M

(G, M, M, I

|M|

where I

|M|

denotes the M × M -identity matrix and G is a nilpotent group.

1.2.2 Basic Deﬁnitions and Some Important Results

Semigroup rings arise naturally as a generalization of group rings w hen

we replace the group by a semigroup.

Deﬁnition 1.2.22. Let S be a semigroup, and R be a ring with identity

1. The semigroup ring RS is the ring of all formal sums α =



s∈S

∈ R, with ﬁnite support supp(α) = {s ∈ S; α

= 0}. We say that α

is the

coeﬃcient of s in α. Two elements α and β in RS are equal if and only if they

have the same coeﬃcients. The sum α + β is the element



s∈S

(α

+ β

)s.

The product αβ is the element



s∈S

s, where γ



x,y∈S

xy=s

, for each

s ∈ S. If R is a commutative ring, then RS is an R-algebra and is called a

semigroup algebra.

Preliminaries

1.2. Semigroup Ring Theory 22

Let S be a semigroup without zero element, and R be a ring. Notice that

 RS × Rθ. In order to “get rid of” the factor Rθ, we will need the

notion of contracted semigroup ring.

Deﬁnition 1.2.23. Let S be a semigroup with a zero element θ, and R be a

ring with identity 1. The contracted semigroup ring R

S of S over R is the

ring RS/Rθ, i.e., the elements of R

S may be identiﬁed with the set of ﬁnite

sums α =



s∈S

s, α

∈ R, s ∈ S \ {θ}, with componentwise addition and

multiplication deﬁned on the R-basis S \ {θ} as

st =



st, if st = θ,

0, if st = θ,

and then extended by distributivity to all e lements. If S is a semigroup

without zero element, we deﬁne the contracted semigroup ring of S over R

as R

S := RS.

Some natural classes of rings may be treated as contracted semigroup

rings and not as (not contracted) semigroup rings.

Example 1.2.24. 1. Let R be a ring, n > 1 be an integer, and let S be

the s emigroup of n × n matrix units with zero (as in Example 1.2.2.1).

Then R

S  M

(R). Notice that, if K is a ﬁeld, then M

(K) is a

simple ring. But a (not contracted) semigroup algebra of a nontrivial

semigroup is never simple, as it contains the augmentation ideal, that

is, the ideal of all elements of the semigroup algebra of which the sum

of the coeﬃcients is zero.

2. Let R be a ring, I and M be nonempty sets and P = (p

m,i

) be an

M × I matrix over R. Consider the set M(R, I, M, P ) of all I × M

matrices over R with ﬁnitely many nonzero entries. For any A = (a

i,m

)

and B = (b

i,m

) ∈ M(R, I, M, P ), addition is deﬁned entrywise and

multiplication is deﬁned as follows:

AB := A ◦ P ◦ B, with ◦ denoting the usual matrix multiplication.

With these operations M(R, I, M, P ) becomes a ring, called a ring of

matrix type over R with sandwich matrix P . If each row and column of

P contains an invertible element of R, then we call M(R, I, M, P )

a Munn ring. Note that if i ∈ I, m ∈ M and p

m,i

is a unit of

R then e := (p

−1

m,i

)

is an idempotent of M(R, I, M, P ) such that

eM(R, I, M, P )e  R. When the sets I and M are ﬁnite, say |I| = n

and |M| = m, we will denote M(R, I, M, P ) simply as M(R, n, m, P ).

Preliminaries

1.2. Semigroup Ring Theory 23

Let G be a group. It is easily veriﬁed that R

, I, M, P ) 

M(RG, I, M, P ) (see Example 1.2.2). If R = ZG, m = n and P is

the n × n identity matrix, then M(R, n, m, P ) = M

(ZG).

3. Let K be a ﬁeld, Ω be any nonempty set, X be the free monoid on

an alphabet {x

; i ∈ Ω}, and I be an ideal in KX generated by an

ideal J of X. We call KX/I a monomial algebra. We have that KX/I

is a K-space with basis the Rees f actor semigroup X/J and, thus,

KX/I  K

(X/J) (see Lemma 1.2.25 below).

Lemma 1.2.25. [4, Lemma 5.12] Let S be a semigroup and I be an ideal in

S. Then R I is an ideal in RS and RS/RI  R

(S/I).

The analogue of Maschke Theorem (Theorem 1.1.14) for semigroup alge-

bras is the following theorem:

Theorem 1.2.26. [30, Theorem 14.24] Let K be a ﬁeld and S a ﬁnite semi-

group. The following conditions are equivalent:

1. KS is a semisimple algebra

2. S

has a series of ideals S

= S

⊃ S

⊃ . . . ⊃ S

⊃ S

n+1

= {θ}, with

each principal factor S

i+1

 M

, n

, P

), a square completely

0-simple matrix semigroup, with G

a ﬁnite group such that char(K) 

| and P

invertible in M

(KG

), for all i.

In the case of inverse semigroups, which are the most useful class of

semigroups in the present work, we have the following special case, that

resembles a lot Maschke Theorem:

Theorem 1.2.27. [4, Theorem 5.26] Let K be a ﬁeld and S a ﬁnite inverse

semigroup. The semigroup algebra KS is semisimple if and only if char(K)

is zero or a prime not dividing the order of any subgroup of S.

The following result shows that rings of matrix type (Example 1.2.24.2)

arise naturally from matrix semigroup rings.

Theorem 1.2.28. [30, Lemma 5.1] Let R be a ring and let S be a matrix

semigroup M

(G, I, M, P ). Then we have that R

S  M(RG, I, M, P ),

where we consider the entries in P as elements of the group ring RG. If S

is completely 0-simple, then M(RG, I, M, P ) is a Munn ring.

Munn rings are very imp ortant in the study of semigroup algebras over a

ﬁnite semigroup, as is shown in the following corollary:

Preliminaries

1.2. Semigroup Ring Theory 24

Corollary 1.2.29. Let S be a ﬁnite semigroup with zero element θ and with a

principal series S = S

⊃ S

⊃ . . . ⊃ S

⊃ S

n+1

= {θ}. Then the semigroup

ring RS has a series of ideals RS = RS

⊃ RS

⊃ . . . ⊃ RS

⊃ RS

n+1

= Rθ,

where each factor RS

/RS

i+1

 R

i+1

) is either a nilpotent ring or a

Munn ring over a group ring.

Example 1.2.30. If S := {e

1,1

, e

1,2

, e

2,2

, θ}, then



Z Z

0 Z



 Z

S. S has

a principal series

S ⊃ {e

2,2

, e

1,2

, θ} ⊃ {e

1,2

, θ} ⊃ {θ},

with Rees factors S/{e

2,2

, e

1,2

, θ}  {e

1,1

}

, {e

2,2

, e

1,2

, θ}/{e

1,2

, θ}  {e

2,2

}

and {e

, θ}/{θ}  {e

1,2

, θ}, a null semigroup.

In order to investigate units in semigroup algebras, it is elucidative to

know when a Munn algebra over a group algebra contains an identity.

Theorem 1.2.31. [30, Corollary 5.26] Let S = M

(G, I, M, P ) be a Rees

matrix semigroup and K be a ﬁeld. The following conditions are equivalent:

1. K

S has an identity;

2. I and M are ﬁnite sets of the same cardinality n and P is an invertible

matrix in M

(KG).

Moreover, if both conditions hold, then K

S  M

(KG) and S is com-

pletely 0-simple.

Preliminaries

Chapter 2

Central Idempotents of Group

Algebras of Finite Nilpotent

Groups

In this chapter, the primitive central idempotents of a semisimple group

algebra of a ﬁnite abelian group over an arbitrary ﬁeld are exhibited. After-

wards, we determine the primitive central idempotents in a complex group

algebra of a ﬁnite nilpotent group (without using group characters).

Let G be a ﬁnite abelian group of order n, and K be a ﬁeld such that

char(K)  n. Consider the abelian group algebra KG. From Theorem 1.1.32,

we know that KG 



K(ζ

), where ζ

are primitive roots of unity whose

orders divide n. Clearly, the primitive idempotents of KG are the inverse

images of each tuple of the form (0, . . . , 0, 1, 0, . . . , 0) under this isomorphism.

We shall describe the primitive idempotents of KG. In particular, we obtain

a description for all cyclic codes (Deﬁnition 2.1.7) over ﬁnite ﬁelds.

For G an arbitrary ﬁnite group, the primitive central idempotents of the

complex group algebra CG are given by the formula

χ(1)

|G|



g∈G

χ(g

−1

)g, where

χ is an irreducible complex character of G and 1 is the identity of G (Theorem

1.1.33). Though theoretically important, this description is not very useful

in practical terms, be cause, with the known methods, the computational

complexity of calculating the character table of a given ﬁnite group grows

exponentially with respect to the order of the group.

Consider now G a ﬁnite nilpotent group. The primitive central idempo-

tents in the rational group algebra QG have been determined at [14], without

making use of the character table of G. We are going to use a similar method

and the abelian case in order to ﬁnd out the primitive central idempotents

in the complex group algebra CG.

Our description allows the construction of the character table of G using a

2.1. Primitive Idempotents of Semisimple Group Algebras of a Finite Abelian Group 26

lattice of subnormal subgroups of G. In particular, our description is helpful

in studying counterexamples to the Isomorphism Problem in group rings ([25,

Chapter 9]).

2.1 Primitive Idempotents of Semisimple

Group Algebras of a Finite Abelian

Group

Let G  C

× . . . × C

be a ﬁnite abelian group of order n, with C

g

; g

= 1 the cyclic group of order n

generated by g

(Structure Theorem

of Finite Abelian Groups), and let K be a ﬁeld such that char(K)  n.

Deﬁne m := lcm(n

, . . . , n

). For i = 1, . . . , s, consider ζ

a primitive

root of unity of order n

in K, an algebraic closure of K. Given an s-tuple

of integers l = (l

, . . . , l

), with 0 ≤ l

≤ n

− 1, deﬁne the polynomial

∈ K(ζ

)[X

, . . . , X

] as:



i=1

−1



=l

− ζ

where ζ

is a primitive root of unity of order m. Notice that P

(ζ

, . . . , ζ

) =

0 if and only if k = l. This polynomial will be useful to describe the primitive

idempotents of KG.

Suppose K is an algebraically closed ﬁeld such that char(K)  n. From

Theorem 1.1.32, it follows that KG  K ⊕ . . . ⊕ K, with n copies of K

on the right side (indeed, in this case, for each divisor d of n, we have that

K(ζ

) = K and, thus, [K(ζ

) : K] = 1, a

[K(ζ

):K]

= n

and



d|n

= n).

The n components of the direct sum K ⊕ . . . ⊕ K will be indexed by the

s-tuple of integers l = (l

, . . . , l

), with 0 ≤ l

≤ n

− 1, in the following

manner: the ﬁrst n

coordinates are the ones having l

= 0, for i = s, and

varying from 0 to n

− 1; the next n

coordinates are the ones having

= 0, for i = s, s − 1, l

s−1

= 1 and l

varying from 0 to n

− 1; the next

coordinates are the ones having l

= 0, for i = s, s − 1, l

s−1

= 2 and l

varying from 0 to n

− 1; and so on.

Lemma 2.1.1. Let G  C

× . . . × C

be the ﬁnite abelian group of order

n, with C

= g

; g

= 1 a cyclic group of order n

generated by g

, and let

K be an algebraically closed ﬁeld such that char(K)  n. The isomorphism

KG  K ⊕ . . . ⊕ K maps

. . . g

→ (. . . , ζ

. . . ζ

, . . .)

0≤k

≤n

−1

Central Idempotents of Group Algebras of Finite Nilpotent Groups

2.1. Primitive Idempotents of Semisimple Group Algebras of a Finite Abelian Group 27

where ζ

∈ K is a primitive root of unity of order n

, for each i = 1, . . . , s.

Proof. We proceed by induction on s, the number of cyclic components of

When s = 1, then G = g; g

= 1 is a cyclic group of order n. De-

ﬁne ζ := ζ

, a primitive root of the unity of order n. Consider v

(1, ζ

, ζ

, . . . , ζ

(n−1)i

) ∈ K ⊕ . . . ⊕ K, for i = 1, . . . , n. We shall see that

; i = 1, . . . , n} is a K-basis for K ⊕ . . . ⊕ K.

Notice that the n×n matrix having vector v

as its i

line is an invertible

Vandermonde matrix. Thus, the n vectors in the set {v

; i = 1, . . . , n} are

linearly independent. Since the K-dimension of K ⊕. . .⊕K is n, we conclude

that {v

; i = 1, . . . , n} is a K-basis for K ⊕ . . . ⊕ K.

Consider the K-linear mapping

φ : KG −→ K ⊕ . . . ⊕ K,

g → (1, ζ, ζ

, . . . , ζ

n−1

Notice that {g

; i = 1, . . . , n} is a K-basis of KG and that φ(g

) = v

. So,

 K ⊕ . . . ⊕ K as K-vector spaces. Clearly, φ is a ring homomorphism,

and hence KG

 K ⊕ . . . ⊕ K as rings too.

Now we consider the case where G  C

× . . . × C

, with s > 1. A ssume

the result holds for any abelian group having s − 1 cyclic components.

Deﬁne G

:= C

× . . . × C

s−1

. We know that

KG  K(G

× C

)  (KG

the isomorphisms being



. . . g

s−1

→



. . . g

s−1

, g



. . . g

s−1

, g

) →



(



,...,x

s−1

. . . g

s−1

where

x := (x

, . . . , x

s−1

, x

), with 0 ≤ x

≤ n

− 1, for each i. By the

induction hypothesis, we have that

 K ⊕ . . . ⊕ K,

. . . g

s−1

→ (. . . , ζ

. . . ζ

s−1

, . . .)

0≤k

≤n

−1

So, it follows that

(KG

 (K ⊕ . . . ⊕ K)C

 KC

⊕ . . . ⊕ KC

Central Idempotents of Group Algebras of Finite Nilpotent Groups

2.1. Primitive Idempotents of Semisimple Group Algebras of a Finite Abelian Group 28

where the last isomorphism is



(



,...,x

s−1

(. . . , ζ

. . . ζ

s−1

, . . .), g

) →

(. . . ,



. . . ζ

s−1

, . . .)

0≤k

≤n

−1

Using the cyclic group case, we have that

⊕ . . . ⊕ KC

 K ⊕ . . . ⊕ K,

(. . . ,



. . . ζ

s−1

, . . .) →

(. . . ,



. . . ζ

s−1

, . . .)

0≤k

≤n

−1

which is the desired result.

Now we describe the primitive idempotents of a semisimple group algebra

of a ﬁnite abelian group over an algebraically closed ﬁeld.

Theorem 2.1.2. Let G  C

× . . . × C

be a ﬁnite abelian group of order n,

with C

= g

; g

= 1 the cyclic group of order n

generated by g

, and let

K be an algebraically closed ﬁeld such that char(K)  n. Then the primitive

idempotents of the abelian group algebra KG are the elements:

, . . . , g

)

(ζ

, . . . , ζ

)

where 0 ≤ l

≤ n

− 1, for i = 1, . . . , s.

Proof. The image of e

,...,g

)

(ζ

,...,ζ

)

under the isomorﬁsm KG

 K ⊕. . .⊕K

speciﬁed in Lemma 2.1.1 is

φ(e

) = (. . . ,

(ζ

, . . . , ζ

)

(ζ

, . . . , ζ

)

, . . .)

0≤k

≤n

−1

Since P

(ζ

, . . . , ζ

) = 0 if and only if k = l, we have that all the coordinates

of φ(e

) are zero, except the one in the position indexed by l, which equals

(ζ

, . . . , ζ

)

(ζ

, . . . , ζ

))

= 1.

Thus, φ(e

) = (0, . . . , 0, 1, 0, . . . , 0), with 1 in the position (l

, . . . , l

), and

hence e

is a primitive idempotent of KG.

Central Idempotents of Group Algebras of Finite Nilpotent Groups

2.1. Primitive Idempotents of Semisimple Group Algebras of a Finite Abelian Group 29

Corollary 2.1.3. With the same notation as in Theorem 2.1.2, suppose that

G = g; g

= 1 is the cyclic group of order n generated by g. Let ζ be a

primitive root of unity of order n. Then the primitive idempotents in KG

are:

n−l

n−1



i=0,i=l

(g − ζ

where 0 ≤ l ≤ n − 1.

Proof. Notice that

(ζ

) =

n−1



i=0

i=l

(ζ

− ζ

) = ζ

n−1



i=1

(1 − ζ

) = ζ

Now, s ince P

(g) =



n−1

i=0

i=l

(g − ζ

), the result follows directly from Theorem

2.1.2.

When the ﬁeld K is not algebraically closed, exhibiting the isomorphism

KG 



K(ζ

), with ζ

primitive roots of unity, is more complicated. In

order to avoid this, we adopt an alternative method and use the algebraically

closed case to get the primitive idempotents:

Theorem 2.1.4. Let G  C

× . . . × C

be a ﬁnite abelian group of order n,

with C

= g

; g

= 1 the cyclic group of order n

generated by g

, and let

K be a ﬁeld such that char(K)  n. Deﬁne m := lcm(n

, . . . , n

). Consider

A := Aut(K(ζ

)|K), the Galois group of the ﬁeld extension K(ζ

)|K. Then,

for a ﬁxed s-tuple of integers l = (l

, . . . , l

), with 0 ≤ l

≤ n

−1, the element

deﬁned below is a primitive idempotent of the abelian group algebra KG:



σ∈A

, . . . , g

)

σ(P

(ζ

, . . . , ζ

))

the sum of all Galois conjugates of P

, . . . , g

)/P

(ζ

, . . . , ζ

), where P

denotes the polynomial in K(ζ

)[X

, . . . , X

] obtained by applying σ to the

coeﬃcients of P

. Furthermore, these are all the primitive idempotents of

KG.

Proof. From the Theorem 2.1.2, it follows that

, . . . , g

)

(ζ

, . . . , ζ

)

Central Idempotents of Group Algebras of Finite Nilpotent Groups

2.1. Primitive Idempotents of Semisimple Group Algebras of a Finite Abelian Group 30

is a primitive idempotent of KG, where K is an algebraic closure of K.

Therefore, f

is also a primitive idempotent in K(ζ

)G. Notice that K(ζ

)

is the minimal ﬁeld extension K

of K such that all the f

’s belong to K

Each σ ∈ A induces a unique automorphism σ

∗

of K(ζ

)G, i.e., for α =



g∈G

g ∈ K(ζ

)G, deﬁne σ

∗

(α) :=



g∈G

σ(α

)g. Thus, σ

∗

) is still a

primitive idempotent of K(ζ

)G and of KG.

Let e be a primitive idempotent in KG. We may write e = f

+ . . . + f

where f

are primitive idempotents in KG. Let K

be the minimal ﬁeld

such that, for each i = 1, . . . , t, f

belongs to K

G. Take τ ∈ Aut(K

|K).

Applying τ

∗

to e, we get e = τ

∗

(e) = τ

∗

) + . . . + τ

∗

) and, by the unique

representation in K

G (Theorem 1.1.11), we have that, for each i = 1, . . . , t,

there exists a j = 1, . . . , t, such that τ

∗

) = f

. Thus, all the f

are Galois-

conjugates of one another. In fact, suppose that (after reordering) f

, . . . , f

are all conjugates of f

and that f

s+1

, . . . , f

, for some 1 < s < t. Then,

deﬁning e



:= f

+ . . . + f

and e



:= f

s+1

+ . . . + f

, we have that e = e



+ e



with e



and e



orthogonal nonzero idempotents in KG, which contradicts the

primitivity of e. So e is exactly the sum of the distinct Galois-conjugates of

a primitive idempotent in KG, and thus, by deﬁnition, e = e

for some l .

Notice that the primitive idempotents e in KG and the set of distinct

’s deﬁned exist in equal number. We conclude that each e

is a primitive

idempotent in KG.

Corollary 2.1.5. With the same notation as in Theorem 2.1.4, suppose

that G = g; g

= 1 is the cyclic group of order n generated by g. Let

ζ be a primitive root of unity of order n. Fix 0 ≤ l ≤ n − 1. Consider

A := Aut(K(ζ)|K), the Galois group of the ﬁeld extension K(ζ)|K. Then

the element e

deﬁned below is a primitive idempotent of the abelian group

algebra KG:



σ∈A

σ(ζ

n−l

)

n−1



i=0

i=l

(g − σ(ζ

)),

Furthermore, these are all the primitive idempotents of KG.

Proof. The proof is a direct application of Theorem 2.1.4 to Corollary 2.1.3.

Example 2.1.6 (Cyclic Codes). In order to make connections between

cyclic codes and the computation of primitive idempotents for semisimple

group algebras of ﬁnite abelian groups, we need some very basic concepts o

coding theory, for which the reference is [37].

Central Idempotents of Group Algebras of Finite Nilpotent Groups

2.2. Primitive Central Idempotents of Complex Group Algebras of a Finite Nilpotent

Group 31

Let F

denote the ﬁnite ﬁeld of q elements, with q a power of a prime

integer p. Consider the n-dimensional vector space F

, whose elements are

n-tuples a = (a

, . . . , a

n−1

Deﬁnition 2.1.7. A code C is a linear subspace of F

. We call n the length

of C and dimC (as a F

-vector space) the dimension of C. A code C is said

to be a cyclic code if its automophism group Aut(C) contains the cyclic shift,

i.e., (c

, c

, . . . , c

n−1

) ∈ C =⇒ (c

, . . . , c

n−1

, c

) ∈ C.

It is common in coding theory to identify F

with the vector space P

polynomials of degree less than n over F

via the correspondence

a = (a

, . . . , a

n−1

) ←→ a(x) = a

+ a

x + . . . + a

n−1

∈ F

[x].

Clearly, a cyclic code C of length n may be identiﬁed with the corresponding

ideal in

[x]

−1)

Let G = g; g

= 1 be the cyclic group of order n generated by g. We

have that F

G 

[x]

−1)

as rings, by taking the morphism g → x. Thus, we

have an immediate connection between group rings and coding theory.

Determining a cyclic code of length n over a ﬁnite ﬁeld K = F

such

that char(K) = p  n corresponds to determining an ideal in the semisimple

group algebra KG, where G is a cyclic group of order n . In this case, all the

ideals of KG are direct summands ⊕

k=1

(KG)e

, where l ≤ n and e

, . . . , e

are the primitive idempotents of KG as determined in Corollary 2.1.5.

2.2 Primitive Central Idempotents of

Complex Group Algebras of a Finite

Nilpotent Group

We shall ﬁrst state some deﬁnitions and results needed on this section.

Deﬁnition 2.2.1. Let R be a ring with identity and let G be a group.

For e a primitive central idempotent of RG, deﬁne the subset G

of G as

:= {g ∈ G; eg = e}.

Deﬁnition 2.2.2. Let G be a ﬁnite group and K be a ﬁeld. Deﬁne the

element ε(G) in KG by

ε(G) :=



1 , if G = {1};



M∈M(G)

(1 −



M) , if G = {1},

where M(G) denotes the set of all minimal nontrivial normal subgroups of

Central Idempotents of Group Algebras of Finite Nilpotent Groups

2.2. Primitive Central Idempotents of Complex Group Algebras of a Finite Nilpotent

Group 32

Remark 2.2.3. Notice that if A and B are subgroups of a group G, with

A ⊆ B, then



B =



B and (1 −



A)(1 −



B) = (1 −



A). So we can redeﬁne

ε(G) as

ε(G) :=



NG

N={1}

(1 −



N),

when G is a nontrivial group. Notice, in particular, that if N is a nontrivial

normal subgroup of G, then (1 −



N)ε(G) = ε(G).

The following lemma is elementary, but extremely useful.

Lemma 2.2.4. Let G be a ﬁnite group, K be a ﬁeld such that char(K)  |G|,

and e be a primitive central idempotent of KG. Then:

1. G

is a normal subgroup of G and e



= e. Thus, e is also a primitive

central idempotent of (KG)



, and, since (KG)



 K(G/G

), the

image e of e in K(G/G

) is a primitive central idempotent of K(G/G

);

2. if N is a normal subgroup of G, then e



N = e if and only if N ⊆ G

;

3. G

= {1} if and only if ε(G)e = e;

4. if H := G

and N is a normal subgroup of G contained in H, then

(G/N)

= H/N, where e denotes the image of e in K(G/N)(we call

attention to this special case: if N is a normal subgroup of G and e is

a primitive central idempotent in K(G/N), then: (G/N)

= {1} if and

only if G

= N, where e denotes the image of e in KG).

Proof. 1. To see that G

is a normal subgroup of G, take h ∈ G and

g ∈ G

. Since e is central, we have that hgh

−1

e = hgeh

−1

= e. So

hgh

−1

∈ G

, and thus G

is normal in G.

By Deﬁnition 2.2.1, we have that





g∈G



g∈G



g∈G

= e.

So e is a primitive central idempotent of (KG)



and e, the image of

e in K(G/G

), is a primitive central idempotent of K(G/G

2. Consider N a normal subgroup of G. Then



N is a central idempotent

of KG. Since e is a primitive central idempotent of KG, we have that



N is either equal to zero or to e.

Central Idempotents of Group Algebras of Finite Nilpotent Groups

2.2. Primitive Central Idempotents of Complex Group Algebras of a Finite Nilpotent

Group 33

Suppose e



N = e. We have that

(KG)e = (KG)



Ne  K(G/N)e.

Thus we have the following diagram

KG −→ K(G/N)

×e

−→ K(G/N)e



−→ (KG)e,

the maps beign

g → g,

g → ge,

ge → g



Ne = ge.

From the diagram, it follows that the ker(KG

×e

−→ (KG)e) = G

and

that N ⊆ ker(KG

×e

−→ (KG)e).

If N ⊆ G

, then



N =



g∈N

|N|



g∈N

|N|



g∈N

|N|

|N|e

|N|

= e.

3. Suppose G

= {1}. If N is a nontrivial normal subgroup of G, then,

by 2., we have that e



N = 0. So, from Deﬁnition 2.2.2, it follows that

ε(G)e =



M∈M(G)

(1 −



M)e =



M∈M(G)

(e −



Me) =



e = e.

Suppose G

= {1}. Then, by Remark 2.2.3, we have that ε(G) =

(1 −



)ε(G) and eε(G) = e(1 −



)ε(G) = 0.

4. This part of the Lemma follows directly from Deﬁnition 2.2.1.

The deﬁnitions and notations needed in the following lemma and theorem

are stated in Deﬁnition 1.1.31.

We need the following technical lemma ([14, Lemma 2.3]):

Lemma 2.2.5. Let G be a ﬁnite group, K be a ﬁeld such that char(K)  |G|,

and g ∈ G. If g

−1

∩ Z(G) = {1}, then G contains a central element z of

prime order so that





z.

Central Idempotents of Group Algebras of Finite Nilpotent Groups

2.2. Primitive Central Idempotents of Complex Group Algebras of a Finite Nilpotent

Group 34

Proof. By assumption there exist h ∈ G and 1 = z ∈ Z(G) so that h

−1

gh =

zg. Hence, for any positive integer n we get, by induction, that h

−n

= z

Replacing z by some power of z, if necessary, we may assume that z has

prime order. It then follows that zC

⊆ C

. So zC

= C

and therefore





z.

Before actually using the lemma, let us just observe that g ∈ C

(G))

implies g

−1

∩ Z(G) = {1}. In fact, suppose that, for g ∈ G, it holds that

−1

∩Z(G) = {1}. Take l ∈ Z

(G). We have that

−1

gl = 1 in G/Z(G)

(because Z

(G)/Z(G) = Z(G/Z(G)), an abelian group); in other words,

there exists z

∈ Z(G) such that g

−1

gl = z

, s o, by hypothesis, z

= 1.

Hence, g ∈ C

(G)).

We need a result from Jespers-Leal-Paques ([14, Proposition 2.1]). This

result is stated in the reference for the group algebra QG, where G is a ﬁnite

nilpotent group. We observe that the proof given in the article is still valid

for the case CG and we include it here for the sake of completeness.

Proposition 2.2.6. Let G be a ﬁnite nilpotent group, e ∈ CG and G

(G)), the centralizer in G of the second center of G. e is a primitive

central idempotent of CG with G

trivial if and only if e =



g∈G

, the sum

of all distinct G-conjugates of e

, with e

a primitive central idempotent of

satisfying



g∈G

((G

)

= {1}.

Proof. Suppose e ∈ CG is a primitive central idempotent with G

= {1}. By

Theorem 1.1.16, we may write e =



g∈G



, with each α

∈ C.

For any g ∈ G with g ∈ C

(G)), by Lemma 2.2.5, there exists a

nontrivial central element w

∈ G of prime order such that





w

.

Hence,

e =



g∈C

(G))





g∈C

(G))





w

.

Because G

= {1}, Lemma 2.2.4 yields that e = eε(G). Notice that

ε(G)



w

 = 0, for, since w

has prime order and is central, w

 ∈ M(G)

and, by Deﬁnition 2.2.2,

ε(G)



w

 =



M∈M(G)

(1 −





w

 =

(



M∈M(G)\{w

}

(1 −



M))(1 −



w

)



w

 =

(



M∈M(G)\{w

}

(1 −



M))(



w

 −



w

) = 0,

i.e., ε(G) ∈ 



; g ∈ C

(G))

, the C-subspace of CG generated by

{



; g ∈ C

(G))}.

Central Idempotents of Group Algebras of Finite Nilpotent Groups

2.2. Primitive Central Idempotents of Complex Group Algebras of a Finite Nilpotent

Group 35

So we get that

e = eε(G) =



g∈C

(G))



· ε(G),

and thus e ∈ 



; g ∈ C

(G))

too.

Notice that G

= C

(G)) is normal in G; so if g ∈ G

and h ∈ C

, then

h ∈ G

. In fact, we have that h = l

−1

gl, for some l ∈ G. Take x ∈ Z

(G). So

−1

xh = z ∈ Z(G) and lx

−1

lxl

−1

gl = lz. Since Z

(G) is normal in

G, we have that y = lx

−1

∈ Z

(G), and thus y

−1

yg = z = 1 (because

g ∈ G

= C

(G))). Hence, x

−1

xh = 1 and h ∈ G

, as desired.

Thus, we have shown that supp(e) ⊆ G

= C

(G)). Notice that e is

not necessarily a primitive central idempotent of CG

. However, it is possible

to write

e = e

+ . . . + e

the sum of all G-conjugates of a primitive central idempotent e

∈ CG

In fact, since e ∈ CG

we may write, by Theorem 1.1.11, e = a

. . . + a

, with {e

, . . . , e

} all the primitive central ide mpotens of CG

and a

∈ CG

. We have, for i = 1, . . . , m, that ee

is either equal to 0 or

to e

, because ee

is an idempotent and e

is a primitive idempotent. Hence,

after possibly reordering the e

’s, we have that e = e

+ . . . + e

, for some

k ≤ m. Now we only have to check that every e

is a G-conjugate of e

. For

g ∈ G, we have that e = e

= e

+ . . . + e

, and by the unique representation

in CG

(Theorem 1.1.13), we have that, for each i = 1, . . . , k, there exists

a j = 1, . . . , k, such that e

= e

. Thus, all the e

are G -c onjugate of one

another. So e is exactly the sum of the distinct G-conjugates of e

Observe that ((G

)

= (G

)

(for g ∈ (G

)

⇐⇒ ge

= e

⇐⇒

−1

= e

⇐⇒ g

−1

∈ (G

)

⇐⇒ g ∈ ((G

)

). Hence, it easily follows

that ∩

i=1

((G

)

= G

= {1}. This proves the necessity of the conditions.

Conversely, suppose that G is a ﬁnite nilpotent group, that e

is a primi-

tive central idempotent of CG

, where G

:= C

(G)), and that

∩

g∈G

((G

)

= {1}. Let e := e

+ . . . + e

be the sum of all dis-

tinct G-conjugates of e

. Clearly, e is a central idempotent of CG and

= ∩

g∈G

((G

)

= {1}.

To see that e is primitive, write e = f

+. . .+f

, a sum of primitive central

idempotents of CG. For any nontrivial central subroup N of G, by Lemma

1.1.17,



N is a central idempotent of CG

(for N ⊆ Z(G) ⊆ C

(G)) = G

so either



= 0 or



= e

. However, the latter is impossible, as it

implies, by Lemma 2.2.4, that N ⊆ (G

)

and thus N ⊆ ∩

g∈G

((G

)

{1}. So we get that



= 0. Recall that, for a nilpotent group, every

Central Idempotents of Group Algebras of Finite Nilpotent Groups

2.2. Primitive Central Idempotents of Complex Group Algebras of a Finite Nilpotent

Group 36

minimal normal subgroup is central ([25, Corollary 1.5.19]). Thus

ε(G)e



M∈M(G)

(1 −



M)e



M∈M(G)

−



) = e

The same argument us ed above to see that



= 0 works to show that



= 0, for any G-conjugate e

of e

. Consequently, ε(G)e = e, and

hence, by the unique representation in CG (Theorem 1.1.13), ε(G)f

= f

Therefore, by Lemma 2.2.4 again, G

= {1}, and, by the ﬁrst part of the

proof, f

∈ CG

. In the ﬁrst part of the proof, we saw that a primitive

central idempotent f

, having G

= {1}, may be written as the sum of all

the distinct G-conjugates of e

. So, by the deﬁnition of e, it follows that

e = f

is a primitive central idempotent of CG.

The following theorem yields an explicit formula for the primitive central

idempotents in CG when G is a ﬁnite nilpotent group.

Theorem 2.2.7. Let G be a ﬁnite nilpotent group. The primitive central

idempotents of the group algebra CG are precisely all elements of the form



g∈G



)

the sum of all distinct G-conjugates of e, where e is an element of CG

such

that e (the image of e in C(G

)) is a primitive central idempotent in

C(G

), having (G

)

= {1}. The groups H

and G

are subgroups

of G satisfying the following properties:

1. H

⊆ H

⊆ . . . ⊆ H

⊆ G

⊆ . . . ⊆ G

⊆ G

= G,

2. for 0 ≤ i ≤ m , H

is a normal subgroup of G

, G

is not abelian for

0 ≤ i < m, and G

is abelian,

3. for 0 ≤ i ≤ m − 1, G

i+1

= C

)),

4. for 1 ≤ i ≤ m,



x∈G

i−1

= H

i−1

Proof. Let us ﬁrst show that the element deﬁned above, satisfying the listed

conditions, is in fact a primitive central idempotent of CG. By condition 2,

is an abelian group. Let f

:= e, a primitive central idempotent in

C(G

), having (G

)

= {1}. Since C(G

)  (CG

)



, we

have that f

:= e



is a primitive central idempotent of (CG

)



and,

thus, it is also a primitive central idempotent of CG

. From (G

)

{1}, we have, by Lemma 2.2.4, that (G

)

= H

; so (G

m−1

)



Central Idempotents of Group Algebras of Finite Nilpotent Groups

2.2. Primitive Central Idempotents of Complex Group Algebras of a Finite Nilpotent

Group 37

m−1

(where f



is any preimage of f

in G

m−1

). Deﬁne f

m−1



g ∈ G

m−1

g

, the sum of all distinct G

m−1

-conjugates of f



Then it is a central idempotent of C(G

m−1

). From condition 4,



g∈G

m−1



m−1

)







g∈G

m−1

)

= {1}.

Conditions 3 and 4 provide the hypotheses for the Proposition 2.2.6, which

yields that f

m−1

is primitive in C(G

m−1

)  (CG

m−1

)



m−1

. Thus,

m−1

:= (



g∈G

m−1

)



m−1

, the image of f

m−1

in (CG

m−1

)



m−1

, is a prim-

itive central idempotent of CG

m−1

. We also have, by condition 2, that:

m−1

= (



g∈G

m−1

)



m−1



g∈G

m−1



m−1

)



g∈G

m−1





m−1

)



g∈G

m−1



)

By induction, we obtain that f



g∈G



)

is a primitive central idem-

potent of CG.

Now, let f

:= d be a primitive central idempotent of CG, where G is a

ﬁnite nilpotent group with nilpotency class c. Then H

:= G

is a normal

subgroup of G

:= G by Lemma 2.2.4, and, since f



= f

, we have that f

a primitive central idempotent of (CG

)



. From (CG)



 C(G

), we

get that f

, the image of f

in C(G

), is a a primitive central idemp otent

of C(G

). Clearly, (G

)

= {1}.

If G

is an abelian group, we know f

from Theorem 2.1.2 and, triv-

ially, d = f



g∈G



)

, because f



= f

and f

is central.

If G

is not an abelian group, let G

be the unique subgroup of G

such that G

= C

)) = G

. Then, by Proposition

2.2.6, we get:



g∈G

where f

is a primitive central idempotent of C(G

), and



x∈G

)

= {1},

with H

the unique subgroup of G

containing H

such that H

)

. So H

is a normal subgroup of G

. Notice that, from the def-

inition of G

, its nilpotency class is at most c − 1. Since C(G

) 

Central Idempotents of Group Algebras of Finite Nilpotent Groups

2.3. Some Questions for Further Investigation 38

(C(G

)



), we have that f

, the image of f

in C(G

), is a prim-

itive central idempotent of C(G

), having (G

)

= {1}. If G

is an abelian group, then we know f

from Theorem 2.1.2. If G

is not

an abelian group, the result follows by induction on the nilpotency class c of

Remark 2.2.8. Notice that the statement and the proof of Theorem 2.2.7

are still valid if we replace C by any algebraically closed ﬁeld K provided that

char(K) does not divide the order of the group. However, historically, the

complex case is of particular interest. So, we decided to state the theorem in

this context.

Remark 2.2.9. Given a ﬁnite nilpotent group G, we know the primitive

central idempotents e

of CG from Theorem 2.2.7. We can then readily com-

pute the irreducible complex characters χ

of G from the formula (Theorem

1.1.33)

(1)

|G|



g∈G

−1

)g,

obtaining the character table of G.

It is known that if two ﬁnite groups G and H have the same character

table, then CG  CH ([25, Theorem 5.22]). Thus our desc ription of the

primitive central idempotents of CG is useful in studying counterexamples

to the Isomorphism Problem in group rings ([25, Chapter 9]).

2.3 Some Questions for Further

Investigation

The results of [14], in which we base our method, were extended and

simpliﬁed in [29], providing an algorithm using only elementary methods

for calculating the primitive central idempotents of QG, when G is a ﬁnite

nilpotent group, among other cases, but not of CG. These improvements

were implemented in a package ([27]) of programs for GAP System, version

4. An experimental comparison of the speed of the algorithm in [27] and

the character method (computing primitive central idempotents from the

character table of the group) was presented in [28] and showed that the ﬁrst is

usually faster. These improvements, however, do not carry on automatically

to the complex case.

Notice that the computational complexity of the method proposed in

Theorem 2.2.7 is still not known. It provides, however, a theoretic alternative

to the usual character method that might be simpler.

Central Idempotents of Group Algebras of Finite Nilpotent Groups

2.3. Some Questions for Further Investigation 39

Hence, calculating the computational complexity of the proposed method

and trying to adapt the ideas in [29] to the complex case may be natural

directions to follow in further studies of the subject.

Central Idempotents of Group Algebras of Finite Nilpotent Groups

Chapter 3

The Normalizer of a Finite

Semigroup and Free Groups in

the Unit Group of an Integral

Semigroup Ring

Given a group G and a commutative ring R, one of the central problems in

Group Ring Theory is deciding to which extent the group ring RG reﬂects the

properties of the group G. More precisely, one might wonder: is it true that

if RG  RH as R-algebras, then G  H? This problem was stated for the

ﬁrst time in Higman’s P h.D. Thesis [13] and is known as the Isomorphism

Problem ([25, Chapter 9]). The answer strongly depends on the ring R

(for instance, from Theorem 1.1.32, it is known that if K is an algebraicaly

closed ﬁeld, G and H are both ﬁnite abelian groups having the same order,

and |G| = |H|  char(K), then KG  KH) and early results seemed to

suggest that, for a given family of groups, it might be possible to obtain

an adequate ﬁeld for which the isomorphism problem would have a positive

answer. However, Dade gave an example in [5] of two nonisomorphic groups

such that their respective group algebras over any ﬁeld are isomorphic.

Of primary importance is the case when R = Z, for if ZG  ZH, then

RG  RH as R-algebras, for any commutative ring R.

The normalizer of the trivial units ±G in the unit group of an integral

group ring ZG of a ﬁnite group G has turned out to be very useful ([15],

[24]) in tackling the isomorphism problem for integral group rings. In par-

ticular, Hertweck’s investigations in [12] have led to a counterexample to the

isomorphism problem.

Quite naturally, the same problem also makes sense for semigroup rings

([25, Chapter 9]). Studying the normalizer for semigroup rings might be

3.1. The Normalizer of a Semigroup 41

helpful in investigating the isomorphism problem for semigroup rings them-

selves, or for group rings, via the connection between these two subjects

provided by partial group rings ([6]). However, in the context of semigroup

rings, very little is known. When we want to investigate this problem for

semigroup rings in a similar way as was done for group rings, we need a

suitable concept of normalizer, as we can no longer speak of trivial units.

In this chapter, we introduce a concept of normalizer of a semigroup S in

the unit group of the integral semigroup ring ZS. We show that this deﬁni-

tion coincides with the normalizer of a group in case of integral group rings

and behaves very much like it in the class of semigroups that are the most

related with groups, namely inverse semigroups. These semigroups have a

natural involution with which we can extend Krempa’s characterization of

the normalizer of the trivial units using the classical involution ([36, Proposi-

tion 9.4]). We will describe the torsion part of the normalizer and study the

double normalizer. Just like in group rings ([8]), the normalizer of a semi-

group contains the ﬁnite conjugacy center and the second center of the unit

group of the integral semigroup ring. We will pose the normalizer problem for

integral semigroup rings and solve it for ﬁnite Malcev nilpotent semigroups

such that the rational semigroup ring is semisimple. Furthermore, just like

in integral group rings ([36, Proposition 9.5]), we get that the normalizer of a

semigroup is a ﬁnite extension of the center of the semigroup ring. All of this

indicates that our concept of normalizer of a semigroup behaves as desired.

Also, using the natural involution on inverse semigroups, we will con-

struct free groups in the unit group of the integral semigroup ring, following

Marciniak and Sehgal [23], using a bicyclic unit and its image under the

involution.

Semigroup rings of inverse semigroups are a wide and interesting class

containing for example matrix rings and partial group rings, for which the

isomorphism problem recently has been investigated (see [6] and [7]).

3.1 The Normalizer of a Semigroup

We start by giving the deﬁnition of the normalizer of a semigroup in

the unit group of its integral semigroup ring. This deﬁnition coincides with

the normalizer of the trivial units in the case of an integral group ring and

behaves very much like it for some semigroups. Many results that hold for

the normalizer of the trivial units in the unit group of an integral group ring

will be extended to the context of integral semigroup rings by means of this

deﬁnition.

Let S be a semigroup. Consider the contracted integral semigroup ring

The Normalizer of a Finite Semigroup and Free Groups in the Unit Group

of an Integral Semigroup Ring

3.1. The Normalizer of a Semigroup 42

with identity (Z

. We denote by U((Z

) the group of units of (Z

Deﬁne the normalizer of ±S as

N(±S) := {u ∈ U((Z

); usu

−1

∈ ±S, for all s ∈ ±S},

which is clearly a semigroup.

Let M

be the subsemigroup of S which is the union of all the maximal

subgroups of S (see Deﬁnition 1.2.6) with a zero θ adjoined, i.e.,

:= (∪

U(E

i,i

)) ∪ {θ},

where the E

i,i

are the idempotents of S. We can hence consider N(±M

)

N(±M

) := {u ∈ U((Z

); usu

−1

∈ ±M

, for all s ∈ ±M

More generally, let H be a subset of U((Z

). Deﬁne the normalizer of

±H in U((Z

) as

N(±H) := {v ∈ U((Z

); v

−1

uv ∈ ±H , for all u ∈ ±H}.

As a special case of this deﬁnition, we call attention to N(N(S).

Given a group G, denote by ∗ the classical involution in the integral group

ring ZG (Deﬁnition 1.1.3), i.e., for α =



g∈G

g ∈ ZG, with α

∈ Z, we

have that α

∗



g∈G

−1

. Denote by U(ZG) the group of units in ZG

and by N

U(ZG)

(±G) the normalizer of the trivial units ±G in U(ZG), i.e.,

U(ZG)

(±G) := {u ∈ U(ZG); ugu

−1

∈ ±G, for all g ∈ ±G}.

Krempa’s characterization of the normalizer in group rings ([36, Propo-

sition 9.4]) states that an element u ∈ U(ZG) belongs to the normalizer

U(ZG)

(±G) if and only if uu

∗

is a central element in U(ZG). So, there

is a close connection between the normalizer and the classical involution.

Therefore, inverse semigroups (Deﬁnition 1.2.6) are the most interesting and

suitable class of semigroups for investigating normalizers in semigroup rings.

For an inve rse semigroup S (Deﬁnition 1.2.6), we c an deﬁne the morphism

 : S → S; s → s



where s



is the unique element in S such that ss



s = s and s



= s



(as

in Deﬁnition 1.2.6). Clearly,  is an involution in S (see [4, Lemma 1.18])

and can be extended linearly to an involution in (Z

, with (1) := 1. We

will often denote (a) by a



, for a ∈ (Z

. Notice that, if a ∈ (Z

and

supp(a) ∈ G, a subgroup of S, then a



= a

∗

. Also, if u ∈ U((Z

), then it

is easy to see that (u



)

−1

= (u

−1

)



The Normalizer of a Finite Semigroup and Free Groups in the Unit Group

of an Integral Semigroup Ring

3.1. The Normalizer of a Semigroup 43

Let S be a ﬁnite inverse semigroup. It is well known ([4, §3.3, Exercise 3])

that every principal factor of S is a Brandt semigroup (i.e., a matrix semi-

group of the form M

(G, n, n, I

), where I

is the n × n identity matrix and

G is a maximal subgroup of S, according to Theorem 1.2.14 and Proposition

1.2.8). Moreover, the rational s emigroup algebra QS is semisimple (Theorem

1.2.27). By Corollary 1.2.29, QS has a series of ideals QS = QS

⊃ QS

⊃

. . . ⊃ QS

⊃ QS

s+1

= Qθ, where S

are ideals in a principal series of S

and S

i+1

 M

, n, n, I

) and QS

/QS

i+1

 Q

i+1

)  M

(QG

with G

, . . . , G

the maximal subgroups of S (up to isomorphism). We have

that

S = Q



× Q



× M

(QG

and

i+1



i+1



× Q

i+1

) 

× M

(QG

for i = 1, . . . , s−1. Thus, (Q

 M

(QG

)⊕. . .⊕M

(QG

). Repeating

the same argument for Z, we have that (Z

 M

(ZG

)⊕. . .⊕M

(ZG

Therefore U((Z

)  GL

(ZG

) ⊕ . . . ⊕ GL

(ZG

). Hence, we can

work “coordinatewise” and we will therefore make the reduction to S a

ﬁnite Brandt semigroup and G a maximal subgroup of S. In this case,

 M

(ZG) and, for a = (a

i,j

) ∈ (Z

, we have that (a

i,j

)



= (a

∗

j,i

Given a group G, we denote by Mon(±G) the group of monomial matrices

over ±G. Notice that for a ∈ Mon(±G) we have that



)

i,j



−1

j,i

, if a

j,i

= 0;

0, if a

j,i

= 0.

Denote by Diag(±G) the subgroup of Mon(±G) consisting of matrices over

±G having nonzero elements only on the diagonal, and denote by Scal(±G)

the matrices of Diag(±G) having the same element on the diagonal.

Remark 3.1.1 (Partial Group Rings). We recall the deﬁnition of partial

group rings and some properties, which will make it possible to relate these

objects with the present work.

Deﬁnition 3.1.2. Given a group G and a ring R with identity, we consider

the semigroup S

generated by the set of symbols {[g]; g ∈ G}, with the

following relations:

1. [s

−1

][s][t] = [s

−1

][st];

2. [s][t][t

−1

] = [st][t

−1

];

The Normalizer of a Finite Semigroup and Free Groups in the Unit Group

of an Integral Semigroup Ring

3.1. The Normalizer of a Semigroup 44

3. [s][e] = [s];

4. [e][s] = [s];

for all s, t ∈ G. The partial group ring R

par

G is the semigroup ring of S

over R, i.e.

par

G := RS

(Notice that relation (4) follows from the previous ones, and thus could be

removed from the list.)

Given a group G, the semigroup S

is an inverse semigroup ([10, Theorem

3.4]) and does not contain a zero element.

Remark 3.1.3. Let S be a ﬁnite semigroup. If all principal factors of S are

of the form M

(G, n, n, P ), with P invertible in M

(ZG), then, by Theorem

1.2.26, (Q

is a semisimple semigroup algebra and (Q

 M

(QG

)⊕

. . . ⊕ M

(QG

), where G

, . . . , G

are the maximal subgroups of S (up to

isomorphism). Again, (Z

 M

(ZG

)⊕. . .⊕M

(ZG

). By considering

the ring isomorphism

f : ZM

(G, n, n, I

) −→ ZM

(G, n, n, P ) deﬁned by A → f(A) := AP

−1

we can work “coordinatewise”, and transport all the results obtained on

inverse semigroups to such a semigroup S.

3.1.1 Characterization of N(±S) and Some Results

Recall that, for any semigroup , all maximal subgroups are ismomorphic

(see Deﬁnition 1.2.6). In case S is a Rees matrix semigroup over a group G,

then all the maximal semigroups are isomorphic to G (see Proposition 1.2.8).

We shall now characterize the normalizer of a semigroup and prove several

interesting properties.

Theorem 3.1.4. Let S be a ﬁnite Brandt semigroup, let M denote the union

of all maximal subgroups of S, and let G be a maximal subgroup of S. We

have that:

1. N(±S) = Scal(N

U(ZG)

(±G))Mon(±G) and N(±S) ⊆ N(±M

);

2. N(±M

) = Mon(N

U(ZG)

(±G));

3. if u ∈ N(±M

) then uu



∈ Diag(Z(U(ZG)));

4. if u ∈ N(±S) then uu



∈ Z(U(Z

)).

The Normalizer of a Finite Semigroup and Free Groups in the Unit Group

of an Integral Semigroup Ring

3.1. The Normalizer of a Semigroup 45

Proof. Being a Brandt semigroup, we have that (see Theorem 1.2.14) S 

(G, n, n, I

) and M

 {s = ge

i,i

∈ M

(G, n, n, I

); g ∈ G, i = 1, . . . , n},

where I

is the n × n identity matrix and e

i,j

are matrix units (Example

1.2.2.1). Let u =



i,j=1

i,j

∈ U((Z

1. For u to be in N(±S), we need that for all s ∈ ±S, there exists a t ∈ ±S,

such that us = tu. Now, s = ge

k,l

and t = he

m,p

, for some g, h ∈ ±G and

then us =



i,k

i,l

and tu =



p,j

m,j

Thus, us = tu means that







m,k

g = hu

p,l

;

i,k

= 0, for all i = m;

p,j

= 0, for all j = l.

When we take k = l (i.e., s = ge

k,k

∈ ±M

), we deduce that







m,k

g = hu

p,k

;

i,k

= 0, for all i = m;

p,j

= 0, for all j = k.

(∗)

Suppose that m = p. Then we get that u

p,k

= 0; hence, u

p,j

= 0 for

1 ≤ j ≤ n, which contradicts that u is a unit.

Therefore, m = p (i.e., t = he

m,m

∈ ±M

) and (∗) becomes







m,k

g = hu

m,k

;

i,k

= 0, for all i = m;

m,j

= 0, for all j = k.

Since g and k are arbitrary, we get that u is a monomial matrix



, with u

∈ N

U(ZG)

(±G).

Now, consider s = ge

k,l

∈ ±S, with k = l (i.e., s ∈ ±S \ ±M

). So

u ∈ N(±S), if

uge

k,l

−1

= u

−1

∈ ±S

(since u

−1



−1

i,m

). As u

∈ N

U(ZG)

(±G), we have u

−1

∈ ±G, for some h ∈ ±G; so u

−1

must be in ±G. Hence, all

entries in u diﬀer up to trivial units. Thus u ∈ Scal(N

U(ZG)

(±G))Mon(±G).

So, N(±S)) ⊆ Scal(N

U(ZG)

(±G))Mon(±G).

Now, to see that Scal(N

U(ZG)

(±G))Mon(±G) ⊆ N(±S), consider u ∈

Scal(N

U(ZG)

(±G))Mon(±G), i.e., u =



i,j=1

i,j

monomial, with v ∈

U(ZG)

(±G) and a

i,j

= 0 or a

i,j

∈ ±G. For s = ge

k,l

∈ ±S, with g ∈

±G, we have that us =



i,k

i,l

= va

, where a

is the only

nonzero element in the k

column of u. Since v ∈ N

U(ZG)

(±G), we have

The Normalizer of a Finite Semigroup and Free Groups in the Unit Group

of an Integral Semigroup Ring

3.1. The Normalizer of a Semigroup 46

that us = va

= hva

, where a

is the only nonzero element

in the l

column of u, for h := va

−1

∈ ±G. So, us = tu, with

t = he

∈ ±S.

2. Notice that the proof that N(±M

) ⊆ Mon(N

U(ZG)

(±G)) is contained

in the proof of 1., as we have seen that for u =



i,j=1

i,j

∈ U((Z

)

and s = ge

k,k

∈ ±M

, us = tu automatically implies that t = he

m,m

∈ ±M

(hence, u ∈ N(±M

)) and u ∈ Mon(N

U(ZG)

(±G)).

To see that Mon(N

U(ZG)

(±G)) ⊆ N(±M

), take u ∈ Mon(N

U(ZG)

(±G)),

i.e., u =



i,j=1

i,j

monomial, with u

i,j

= 0 or u

i,j

∈ N

U(ZG)

(±G), and

s = ge

k,k

∈ ±M

, with g ∈ ±G. We have that us =



i,k

= u

where u

is the only nonzero element in the k

column of u. Since u

∈

U(ZG)

(±G), we have that us = u

= hu

, for some h ∈ ±G.

So, us = tu, with t = he

∈ ±M

3. Take u ∈ N(±M

). From the ﬁrst part of the proof, we know that

u ∈ Mon(N

U(ZG)

(±G)), i.e., u = (u

i,j

) monomial, with u

i,j

= 0 or u

i,j

∈

U(ZG)

(±G). So u



= (u

∗

j,i

). By computing uu



, we get a diagonal matrix

with u

i,j

∗

i,j

in the (i, i) position, when u

i,j

= 0. From [36, Proposition 9.4],

we know that u

i,j

∗

i,j

∈ Z(U(ZG)). So, we have the desired result.

4. Following the same lines of 3., we get the analogue result for N(±S).

We get, as an easy but important consequence, that an element of the

normalizer commutes with its image under the involution .

Corollary 3.1.5. Let S be a ﬁnite Brandt semigroup, l et M denote the

union of all maximal subgroups of S, and let G be a maximal subgroup of S.

If u ∈ N(±M

), then uu



= u



Proof. Take u ∈ N(±M

), say u = (u

i,j

) monomial, with u

i,j

= 0 or u

i,j

∈

U(ZG)

(±G). From Theorem 3.1.4, (3), uu



∈ Diag(Z(U(ZG))), i.e., for

each i, j such that u

i,j

= 0, we have that u

i,j

∗

i,j

∈ Z(U(ZG)). So, for all i, j,

i,j

∗

i,j

−1

i,j

∗

i,j

)

−1

= u

−1

i,j

∗

i,j

)(u

∗

i,j

)

−1

= 1.

Thus u

i,j

∗

i,j

= u

∗

i,j

, for all i, j, and we have that uu



= u



For group rings, we know that the only unitary units (Deﬁnition 1.1.30)

for the classical involution are the trivial units (Proposition 1.1.34). We now

describe the analogous “-unitary elements”.

Theorem 3.1.6. Let S be a ﬁnite Brandt semigroup and G a maximal sub-

group of S. For v ∈ (Z

, vv



= 1 if and only if v ∈ Mon(±G).

The Normalizer of a Finite Semigroup and Free Groups in the Unit Group

of an Integral Semigroup Ring

3.1. The Normalizer of a Semigroup 47

Proof. This proof is very similar to the proof of Proposition 1.1.34.

Being a Brandt semigroup, we have that (see Theorem 1.2.14) S 

(G, n, n, I

) and (Z

 M

(ZG) (see Example 1.2.24).

Let v = (v

i,j

) ∈ (Z

, with v

i,j



g∈G

i,j

(g)g ∈ ZG. From vv



= 1 =

, we get in particular that, for all i,



i,j

∗

i,j



(



g∈G

i,j

(g)

) = 1.

Hence, for each i, exactly one v

i,j

(g) = 1, i.e., for each i, exactly one v

i,j

a trivial unit and all the other ones are zero. Again, from vv



= I

, we can

deduce relations between the rows and columns from which it follows that

v ∈ Mon(±G).

Cleary, if v ∈ M on(±G), then v



= v

−1

Next, we prove some results for the normalizer in semigroup rings, that

were proved for group rings in [18], [22] , [24].

The following Lemma is, in a certain sense, the semigroup ring analogue

of a famous corollary of Berman–Higman Lemma for group rings (Corollary

1.1.36). It will be very helpful on many of the results to be proved, and it

has an interesting immediate Corollary.

Lemma 3.1.7. Let S be a ﬁnite Brandt semigroup and G a maximal subgroup

of S. If c ∈ D iag(Z(U(ZG))) and c is a torsion unit, then c ∈ Diag(±Z(G)).

Proof. Take c ∈ Diag(Z(U(ZG))) a torsion unit, say c = (c

i,j

), with c

i,j

= 0,

if i = j, and c

i,i

∈ Z(U(ZG)) torsion units in ZG. We have from Corollary

1.1.37, that, for all i, c

i,i

∈ ±Z(G), i.e., c ∈ D iag(±Z(G)).

Corollary 3.1.8. Let S be a ﬁnite Brandt semigroup, l et M denote the

union of all maximal subgroups of S, and let G be a maximal subgroup of S.

If u ∈ N(±M

), then either uu



has inﬁnite order, or uu



= 1 and, in this

case, u ∈ Mon(±G).

Proof. Take u ∈ N(±M

), say u = (u

i,j

) monomial, with u

i,j

= 0 or u

i,j



g∈G

i,j

(g)g ∈ N

U(ZG)

(±G). From Theorem 3.1.4, (3), we know that uu



c ∈ D iag(Z(U(ZG))). Suppose c is a torsion unit. We have, from Lemma

3.1.7, that c = (c

i,j

) ∈ Diag(±Z(G)).

For each i, c

i,i

= u

i,j

∗

i,j

is a trivial unit with c

i,i

(1) =



g∈G

i,j

(g)

= 0.

It follows that, for each i, c

i,i

= 1, i.e., uu



= 1. From Theorem 3.1.6, it now

follows that u ∈ Mon(±G).

Denote by T (N(±M

)) the set of torsion units in N(±M

). We can prove

that the elements in Mon(±G) are the only torsion elements in N(±M

) and

in N(±S).

The Normalizer of a Finite Semigroup and Free Groups in the Unit Group

of an Integral Semigroup Ring

3.1. The Normalizer of a Semigroup 48

Proposition 3.1.9. Let S be a ﬁnite Brandt semigroup, let M denote the

union of all maximal subgroups of S, and let G be a maximal subgroup of S.

We have that T (N(±M

)) = Mon(±G) = T (N(±S)).

Proof. Take u ∈ T (N(±M

)). From Corollary 3.1.5, we know that uu



a torsion unit and, from Corollary 3.1.8, we get uu



= 1, and thus u ∈

Mon(±G).

The other inclusion is obvious.

The proof is similar for T (N(±S)).

We can now characterize when the normalizer coincides with the torsion

units.

Proposition 3.1.10. Let S be a ﬁnite Brandt semigroup, let M denote the

union of all maximal subgroups of S, and let G be a maximal subgroup of S.

The following are equivalent:

1. N(±M

) = T(N(±M

));

2. N(±S) = T (N(±S));

3. Z(U(ZG)) = ±Z(G).

Proof. From 1. to 2., it is obvious, using that N(±S) ⊆ N(±M

) and

Proposition 3.1.9.

Now, suppose 2. is true. Since Z(U(Z

))  Scal(Z(U(ZG))) is con-

tained in N(±S) = T (N(±S)), we have by Lemma 3.1.7 that Z(U(ZG)) =

±Z(G).

If 3. holds, then from Theorem 3.1.4, (3) it follows that for all u ∈

N(±M

), uu



∈ Diag(±Z(G)); in particular, uu



has ﬁnite order. Thus,

from Corollary 3.1.8, u ∈ Mon(±G) ⊆ T (N(±M

)). The other inclusion is

always true.

For a ﬁnite group G, a well known result due to Krempa [36, Proposition

9.4] states that: for u ∈ U(ZG), we have that u ∈ N

U(ZG)

(±G) ⇐⇒ uu

∗

∈

Z(U(ZG)). We will prove a similar result for N(±Mon(G)), but ﬁrst we

need the following lemma.

Lemma 3.1.11. Let S be a ﬁnite Brandt semigroup, S = M({1}, 2, 2, I

)

and G a maximal subgroup of S. If t ∈ U((Z

) is such that ta = at, for

all a ∈ Mon(±G), then t ∈ Z(U(Z

)).

The Normalizer of a Finite Semigroup and Free Groups in the Unit Group

of an Integral Semigroup Ring

3.1. The Normalizer of a Semigroup 49

Proof. We have that S  M

(G, n, n, I

), with I

the n × n identity matrix.

We divide the proof in two parts.

Part 1: If t ∈ (Z

and ta = at, for all a ∈ Mon(±G), then there exist

α ∈ Z(ZG) and β ∈ Z



G such that t = (t

i,j

), with t

i,j



α, if i = j

β, if i = j

We prove this by induction on n.

For n = 1 the result is obvious.

So, assume the result is valid for n − 1 and we will verify it for n. Take

t ∈ (Z

 M

(ZG) such that ta = at, for all a ∈ Mon

(±G) (i.e., the

monomial n × n matrices over ±G). So, t =







1,1

. . . t

1,n



n,1







, where



∈ M

n−1

(ZG) and it is easy to see that t



b = bt



, for all b ∈ Mon

n−1

(±G),

as e

1,1

+ b ∈ Mon

(±G). Hence by the induction hypothesis, there exist

α ∈ Z(ZG) and β ∈ Z



G such that t



i,j



α, if i = j

β, if i = j

Take a =



i,i

∈ Mon(±G). We have that ta =



i,j

and

at =



i,j

. Thus, for every i and j,

i,j

= g

i,j

.(∗∗)

This means, in particular, that t

i,i

= g

i,i

and, since this is valid for an

arbitrary g

∈ G, we have that t

i,i

∈ Z(ZG), for every i. For i = j, we get

that t

i,j

∈ Z



Now, take a =



i,n+1−i

∈ Mon(±G). Then ta =



i,n+1−j

n+1−j

i,j

and at =



n+1−i,j

i,j

. Thus, for every i and j, t

i,n+1−j

n+1−j

= g

n+1−i,j

So, for all j, g

n,j

= t

1,n+1−j

n+1−j

= g

1,n+1−j

, the last equality following

from (∗∗). Hence, for all j, t

1,n+1−j

= t

n,j

, and, by the induction hypoth-

esis, for j = 1, n, we have that t

1,j

= t

n,n+1−j

= β. Similarly, for all i,

n+1−i,1

= t

i,n

= g

i,n

, the last equality following from (∗∗). Hence, for

all i, t

(n+1−i),1

= t

i,n

, and, by the induction hypothesis, for i = 1, n, we have

that t

i,1

= t

(n+1−i),n

= β. Also, we get t

1,1

= t

n,n

= t

i,i

= α, by the induction

hypothesis again.

So, the only entries in t that are still unknown are t

1,n

and t

n,1

Consider a = g

1,1



i=2

i,n+2−i

∈ Mon(±G). Then we have that

ta =



i=1

i,1



j=2

i,n+2−j

n+2−j

i,j

) and at =



j=1

1,j



i=2

n+2−i,j

i,j

). Thus, g

1,n

= t

1,2

= g

1,2

, the last equality following

from (∗∗). Hence, t

1,n

= t

1,2

= β, by the induction hypothesis.

Finally, consider a = g

n,n



n−1

i=1

ei, n − i ∈ Mon(±G). We have

that ta =



i=1

i,n



n−1

j=1

i,n−j

n−j

i,j

) and at =



j=1

n,j

The Normalizer of a Finite Semigroup and Free Groups in the Unit Group

of an Integral Semigroup Ring

3.1. The Normalizer of a Semigroup 50



n−1

i=1

n−i,j

i,j

). Thus, g

n,1

= t

n,n−1

n−1

= g

n,n−1

, the last equality

following from (∗∗). Hence, t

n,1

= t

n,n−1

= β, by the induction hypothesis.

And this concludes the proof of the ﬁrst part of the Lemma.

Part 2: If t ∈ U((Z

) is such that t = (t

i,j

), with t

i,j



α, if i = j

β, if i = j

for some α ∈ Z(ZG) and β = b



G ∈ Z



G, then t ∈ Z(U((Z

)).

The case n = 1 is trivial.

For the rest of the proof, we assume n > 2 or |G| > 1.

Take t as in the statement above. Notice that, by performing elementary

operations on t, det(t) is multiplied by ±1 (since we are working on integral

matrices). Then, by bringing t to its row-echelon form, it is easy to see

that det(t) = ±(α − β)

n−1

(α + (n − 1)β). Since t is a unit, we have that

det(t) ∈ U(ZG). Consider ε the augmentation mapping in ZG. If we take

ε(det(t)), we get that it is in Z and it is a unit; so

ε(det(t)) = ±(ε(α) − b|G|)

n−1

(ε(α) + (n − 1)b|G|) = ±1.

We claim that the only integer solutions to these equations are ε(α) = ±1

and b = 0.

Let us ﬁrst take a look at (ε(α) − b|G|)

n−1

(ε(α) + (n − 1)b|G|) = 1. So,

either (ε(α)−b|G|)

n−1

= 1 and (ε(α)+(n−1)b|G|) = 1, or (ε(α)−b|G|)

n−1

−1 and (ε(α) + (n − 1)b|G|) = −1. If n is odd, the second option can never

occur, and the only possible integer solutions for the ﬁrst case are ε(α) = ±1

and b = 0 (since the case n = 1 has already been dealt with and is excluded).

If n is even, we have the desired result immediately.

Analyzing equation (ε(α ) − b|G|)

n−1

(ε(α) + (n − 1)b|G|) = −1, we get

either (ε(α)−b|G|)

n−1

= −1 and (ε(α )+(n−1)b|G|) = 1, or (ε(α)−b|G|)

n−1

1 and (ε(α) + (n − 1)b|G|) = −1. If n is odd, the ﬁrst option is not possible,

and the only existing integer solutions for the second case are ε(α) = ±1 and

b = 0. If n is even, then the only possible integer solutions are the desired

ones (since we have already dealt with the cases n = 2 and G trivial, and

n = 1) .

Because t is a unit and a diagonal matrix with α on all nonzero entries,

we have that α ∈ Z(U(ZG)), which ﬁnishes the proof.

Theorem 3.1.12. Let S be a ﬁnite Brandt semigroup, let G be a maximal

subgroup, and let u ∈ U((Z

). We have that u ∈ N(Mon(±G)) if and

only if uu



∈ Z(U((Z

)).

Proof. We examine separately the case where S = M({1}, 2, 2, I), with I the

2 × 2 identity matrix. We have that (Z

= M

(Z) and Z(U((Z

)) =

The Normalizer of a Finite Semigroup and Free Groups in the Unit Group

of an Integral Semigroup Ring

3.1. The Normalizer of a Semigroup 51

±I

. By means of elementary matrix computations, we see that

N(Mon(±{1})) = {



±1 0

0 ±1





0 ±1

±1 0



and u ∈ N(Mon(±G)) implies uu



= I

. Now, if uu



= ±I

, for u ∈

U(M

(Z)), then, making elementary matrix computations, we get that u =



±1 0

0 ±1



or u =



0 ±1

±1 0



. In either case, u ∈ N(Mon({±1})).

For the rest of the proof, we assume S = M({1}, 2, 2, I).

Take u ∈ U((Z

) and a ∈ Mon(±G). Assume u ∈ N(Mon(±G)).

Then b := uau

−1

∈ Mon(±G). Hence, applying  to both sides, b



= b

−1

)



−1



. Taking inverses in this equality, b = (u



)

−1



. Thus,

a = u



b(u



)

−1

So,



ua(u



−1

= u



uau

−1



)

−1

= u



b(u



)

−1

= u



)

−1



)

−1

= a.

We have proved that u



u commutes with elements from Mon(±G). By

Lemma 3.1.11, it follows that u



u ∈ Z(U(Z

)). So u



u = (u



)

−1



∈ Z(U(Z

)).

Now, suppose uu



∈ Z(U((Z

)). Then uu



= u



u and (u



−1

)



∈ Z(U((Z

)). We want to show that uau

−1

∈ Mon(±G), for

all a ∈ Mon(±G). We have that

(uau

−1

)(uau

−1

)



= ua(u

−1

)



−1



= uu

−1

)



= 1.

So, by Theorem 3.1.6, uau

−1

∈ Mon(±G), as desired.

By Theorem 3.1.4, (4) we get the following Corollary:

Corollary 3.1.13. Let S be a ﬁnite Brandt semigroup and let G be a maximal

subgroup of S. Then N(±S) ⊆ N(Mon(±G)).

The reverse inclusion remains an open and interesting problem.

When the central units of ZG are trivial, the problem is solved.

Proposition 3.1.14. Let S be a ﬁnite Brandt semigroup and let G be a

maximal subgroup of S. We have that N(±S) = Mon(±G) = N(Mon(±G))

if and only if Z(U(ZG)) = ±Z(G).

The Normalizer of a Finite Semigroup and Free Groups in the Unit Group

of an Integral Semigroup Ring

3.1. The Normalizer of a Semigroup 52

Proof. Suppose N(±S) = Mon(±G) = N(Mon(±G)). As Z(U(Z

)) 

Scal(Z(U(ZG))) is contained in N(±S) = Mon(±G), we have by Lemma

3.1.7 that Z(U(ZG)) = ±Z(G).

In case Z(U(ZG)) = ±Z(G), then from Theorem 3.1.4, (4) it follows

that for all u ∈ N(±S), uu



∈ Scal(±Z(G)); in particular, uu



has ﬁnite

order. Thus, from Corollary 3.1.8, u ∈ Mon(±G). The other inclusion is

always true. Take u = (u

i,j

) ∈ N(Mon(±G)) with u

i,j



g∈G

i,j

(g)g ∈

ZG. We have, by Theorem 3.1.12, that uu



= (c

i,j

) ∈ Z(U((Z

)) =

Scal(±Z(G)). For each i, c

i,i



i,j

∗

i,j

is a trivial unit with c

i,i

(1) =



(



g∈G

i,j

(g)

) = 0. It follows that, for each i, c

i,i

= 1, i.e., uu



= 1, and

thus u ∈ Mon(±G) by Theorem 3.1.6. The other inclusion is obvious.

Next, we prove a result for the double normalizer in semigroup rings, that

was shown for group rings by Li in [18].

Theorem 3.1.15. Let G be a ﬁnite group. Then N(N(Mon(±G))) =

N(Mon(±G)).

Proof. Take v ∈ N(N(Mon(±G)) and a ∈ Mon(±G). Clearly, Mon(±G) ⊆

N(Mon(±G)). So, from Theorem 3.1.12, v

−1

av(v

−1

av)



∈ Z(U(Z

)) 

Scal(Z(U(ZG))). Since u := v

−1

av and u



= (v

−1

av)



are commuting torsion

units, their product is also a torsion unit. Thus, by Lemma 3.1.7 we have

that c := uu



∈ Scal(±Z(G)).

Now let u = (u

i,j

), with u

i,j



g∈G

i,j

(g)g ∈ ZG. For each i, c

i,i



i,j

∗

i,j

is a trivial unit, with c

i,i

(1) =



(



g∈G

i,j

(g)

) = 0. So, it

follows that, for each i, c

i,i

= 1, i.e., uu



= 1. Thus a(vv



) = (vv



)a, for



= a

−1

. From Lemma 3.1.11, this means that vv



∈ Z(U(Z

S))). Hence,

v ∈ N(Mon(±G)), as desired.

The other inclusion is obvious.

Remark 3.1.16. The deﬁnitions of the hypercenter Z

∞

(G) and the ﬁnite

conjugacy center Φ(G) of a group G are stated in Deﬁnition 1.1.31.

The hypercenter and the ﬁnite conjugacy center of the unit group of an

integral group ring and of an integral semigroup ring have been given special

attention in recent years.

It is a well known result that, if G is a ﬁnite group, then Z

∞

(ZG)) =

(ZG)) (see [1]) and that Z

∞

(ZG)) ⊆ N

(ZG)

(G) (see [20] and [19]),

where U

(ZG) stands for the normalized units of ZG, i.e., the units in ZG

having augmentation 1.

In Corollary 5.2 and 5.3 in [8], a description is given for the ﬁnite conju-

gacy center Φ(U((Z

)) and second center Z

(U((Z

)) of the unit group

The Normalizer of a Finite Semigroup and Free Groups in the Unit Group

of an Integral Semigroup Ring

3.1. The Normalizer of a Semigroup 53

of an integral semigroup ring of a ﬁnite semigroup S such that QS is semisim-

ple. The result shows that both groups equal the hypercenter Z

∞

(U((Z

))

and are central if and only if S has no principal factors which are so called

∗

-groups. A torsion group G is said to be a Q

∗

-group if G has an abelian

normal subgroup A of index 2 which has an element a of order 4 such that,

for all h ∈ A and all g ∈ G \ A, g

= a

and g

−1

hg = h

−1

; or, equiva-

lently, U(ZG) contains an abelian periodic normal subgroup H ⊆ G such

that H ⊆ Z(G) (see [2], [3]).

From the description of N(±S) we gave, it follows that the ﬁnite conju-

gacy center Φ(U((Z

)) and the second center Z

(U((Z

)) (hypercenter)

are always contained in N(±S), similarly to the group ring case, i.e.,

(U((Z

)) = Z

∞

(U((Z

)) = Φ(U((Z

)) ⊆ N(±S).

3.1.2 The Normalizer Problem for Semigroup Rings

The normalizer problem for group rings [36, Problem 43] asks whether

U(ZG)

(±G) = Z(U(ZG))(±G),

where G is a group. It has a positive answer for many classes of ﬁnite groups,

among which ﬁnite nilpotent groups [36, Corollary 9.2]. One can now state

this problem in the setting of semigroup rings: is it true that

N(±S) = Z(U((Z

))Mon(±G)?

In the case of a Malcev nilpotent semigroup S such that QS is semisimple

we have a positive answer to the analogous problem in semigroup rings.

The only Malcev nilpotent completely 0-simple semigroups are the Brandt

semigroups over a nilpotent group ([16, Lemma 2.1]). As observed before,

we can assume that the principal series of S has only one Rees factor.

Theorem 3.1.17. Let S be a ﬁnite Brandt semigroup over a ﬁnite nilpotent

group G and let M denote the union of all maximal subgroups of S. Then

N(±M

) = Mon(Z(U(ZG))(±G)) and N(±S) = Z(U(Z

)Mon(±G).

Proof. From Theorem 3.1.4, (2), N(±M

) = Mon(N

U(ZG)

(±G)). Because S

is Malcev nilpotent, we have that G is a ﬁnite nilpotent group (see Example

1.2.21). Thus, the normalizer problem for ZG has a positive answer and

U(ZG)

(±G) = Z(U(ZG))(±G). So,

N(±M

) = Mon(N

U(ZG)

(±G)) = Mon(Z(U(ZG))(±G)).

Following the same lines, we get the result for N(±S).

The Normalizer of a Finite Semigroup and Free Groups in the Unit Group

of an Integral Semigroup Ring

3.2. Free Groups generated by Bicyclic Units 54

In general, the normalizer problem does not hold for group rings, though

the normalizer is known up to ﬁnite index [36, Proposition 9.5]. We get the

same result in semigroup rings.

Proposition 3.1.18. Let S be a ﬁnite Brandt semigroup, let M denote the

union of all maximal subgroups of S, and let G be a maximal subgroup of S.

If u ∈ N(±M

), then u

∈ Diag(Z(U(ZG)))Mon(±G), and if u ∈ N(±S),

then u

∈ Z(U(Z

))Mon(±G).

Proof. Take u ∈ N(±M

). Deﬁne v := u



−1

, then vv



= u



−1

)



u =



−1

u. From Corollary 3.1.5, we have vv



= u



(uu



)

−1

u = 1. Hence by

Theorem 3.1.6 we have that v ∈ Mon(±G). So, u



= vu and uu



= u



u =

∈ Diag(Z(U(ZG))) by Theorem 3.1.4, (3).

The same reasoning gives the result for u ∈ N(±S).

Corollary 3.1.19. Let S be a ﬁnite Brandt semigroup and G a maximal

subgroup of S. Then

N(±S)

Z((U(Z

))Mon(±G)

is an elementary abelian 2-group.

3.2 Free Groups generated by Bicyclic Units

By Hartley and Pickel [11], we know that there are free groups contained

in the unit group of an integral semigroup ring. Marciniak and Sehgal [23]

constructed a free group of rank 2 in the unit group of an integral group ring

using a nontrivial bicyclic unit (Deﬁnition 1.1.28) and its image under the

classical involution (Deﬁnition 1.1.3).

Since a Brandt semigroup S has the involution  (see Section 3.1 and Def-

inition 1.2.11), we can investigate the same problem for a nontrivial bicyclic

unit of (Z

Theorem 3.2.1. Let S be a Brandt semigroup. Take s ∈ S such that the

cyclic semigroup s is a group of order n, and t ∈ S such that u

s,t

= 1 +

(1 − s)ts is a nontrivial bicyclic unit in (Z

. Then:

1. if st is in a maximal subgroup G of S and if s is not normal in t,

then u

s,t

, (u

s,t

)



 is a free subgroup of U((Z

);

2. if st = 0 and o(s) ≥ 2, then u

s,t

, (u

s,t

)



 is a free subgroup of

U((Z

);

The Normalizer of a Finite Semigroup and Free Groups in the Unit Group

of an Integral Semigroup Ring

3.3. Some Questions for Further Investigation 55

3. if st = 0 and o(s) = 1, then u

s,t

, (u

s,t

)



 is not a f ree subgroup of

U((Z

Proof. Recall that, being a Brandt semigroup, we have that (see Theorem

1.2.14) S  M

(G, n, n, I

) and M

 {s = ge

i,i

∈ M

(G, n, n, I

); g ∈

G, i = 1, . . . , n}, where I

is the n × n identity matrix and e

i,j

are matrix

units (Example 1.2.2.1).

Since s generates a subgroup of order n in S, it is of the form s = ge

i,i

for g ∈ G, i = 1, . . . , n. Notice that in order for u

s,t

to be nontrivial we must

have t = he

j,i

, for h ∈ G, j = 1, . . . , n.

1. If st is in a maximal subgroup G of S, then we must have j = i. So

s,t

= u

g,h

i,i

and, since s is not normal in t (i.e., h ∈ N

(g), u

s,t

is a

nontrivial bicyclic unit in (Z

and u

g,h

is a nontrivial bicyclic unit in ZG.

So (u

s,t

)



= u

∗

g,h

i,i

. Thus, Marciniak and Sehgal’s main result in [23] gives

us that u

s,t

, (u

s,t

)



 is a free subgroup of U((Z

2. If st = 0, this means that j = i. Then u

s,t

= 1 + hge

j,i

and (u

s,t

)



1 + gh

−1

i,j

When o(g) ≥ 2, consider the ring homomorphism

φ : (Z

−→ M

(Z) given by α = (α

i,j

) ∈ (Z

→ (ε(α

i,j

)) ∈ M

(Z),

where ε is the augmentation mapping in ZG. By Sanov’s Theorem ([25, The-

orem 10.1.3]) applied to the 2 × 2 nonzero submatrix of φ(u

s,t

) and φ(u

s,t

)



we obtain the result.

3. Since st = 0, we have that j = i and u

s,t

= 1+hge

j,i

, (u

s,t

)



= 1+gh

−1

i,j

When g = 1, then u

s,t

= 1+he

j,i

, (u

s,t

)



= 1+h

−1

i,j

and ((u

s,t

)



)

−1

= 1−

−1

i,j

. Thus, by pe rforming elementary matrix computations, u

s,t

((u

s,t

)



)

−1

is of order 6. Hence, u

s,t

, (u

s,t

)



 is not a free group.

3.3 Some Questions for Further

Investigation

So far, in this chapter, we have always considered semigroups for which

the rational semigroup algebra is semisimple. It remains open what N(±S) is

in case there is a Jacobson radical (see Deﬁnition 1.1.4 and Theorem 1.1.10).

The following example indicates that N(±S) might always be central.

Example 3.3.1. Let G be a ﬁnite group. Deﬁne S := {Ge

1,2

, Ge

1,3

, G

2,3

where e

i,j

are 3 × 3 matrix units (Example 1.2.2.1). Then T := (S

)

is a

The Normalizer of a Finite Semigroup and Free Groups in the Unit Group

of an Integral Semigroup Ring

3.3. Some Questions for Further Investigation 56

ﬁnite semigroup with zero element and identity and

Γ := (Z

T )

= Z

T =











z ZG ZG

0 z ZG

0 0 z





; z ∈ Z







is an integral semigroup ring, having Jacobson radical J (Γ) = ZGe

1,2

ZGe

1,3

+ ZGe

2,3

. Observe that U(Γ) = 1 + J (Γ).

Let us now compute N(±T ). Take u = ±(



i=1

i,i

) + te

1,2

+ ve

1,3

+ xe

2,3

where t, v, x ∈ ZG. In order to normalize ge

1,2

, with g ∈ ±G, it follows that

x = 0. Note that u already normalizes ge

1,3

, and that ge

2,3

is normalized by

u if and only if t = 0.

Hence,

N(±T ) = {±(



i=1

i,i

) + ve

1,3

; v ∈ ZG} = Z(U(Γ)).

As it has been said in the beginning of this chapter, the study of the

normalizer for semigroup rings, in analogy to what happens in group rings,

might be e lucidative in tackling the isomorphism problem. However, this

connection still has to be made in the context of semigroup rings, and this

is a natural direction for further studies on the subject.

It should also be interesting to investigate the usefulness of the results

obtained to the isomorphism problem for partial group rings, for which it may

be necessary to further explore the structure of the semigroup S

associated

to a given group G.

The Normalizer of a Finite Semigroup and Free Groups in the Unit Group

of an Integral Semigroup Ring

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Bibliography

Index

I × M matrix over G

, 15

Rees matrix, 15

zero matrix, 15

regular matrix, 18

augmentation ideal, 1

∆

(G, N), 1

augmentation mapping, 1

, 1

character table, 7, 12, 25

code, 31

cyclic, 31

cyclic shift, 31

dimension, 31

length, 31

ﬁeld

perfect, 3

splitting, 8

group, 14

center, 9

∗

-group, 53

abelian, 14

central height, 9

centralizer, 9

commutative, 14

commutator, 9

FC center, 9

hypercenter, 9

subgroup, 16

upper central series, 9

with zero, 15

group algebra, 1



H, 32

abelian group

central idempotents, 25, 28,

polynomial P

, 26

complex ﬁeld

central idempotents, 25, 36

Isomorphism Problem, 26, 38,

nilpotent group

central idempotents, 25, 36

primitive central idempotent

, 31

ε(G), 31

group character, 6

irreducible, 6

regular, 6

group representation, 5

degree, 5

equivalent, 5

irreducible, 5

matrix representation, 5

regular, 6

group ring, 1



H, 4

a, 4



H, 4, 5

We dderburn–Artin decomposi-

tion, 7, 10

bicyclic unit, 8

class sums, 4

coeﬃcient of g in α, 1

involution, 2

classical, 2

Isomorphism Problem, 26, 38,

normalized unit, 52

primitive central idempotent

, 31

primitive central idempotents,

product of elements, 1

sum of elements, 1

support of an element, 1

trivial unit, 8

unitary unit, 9

We dderburn–Artin decomposi-

tion, 25, 26

Maschke Theorem, 4

matrix unit, 16, 22

module

completely reducible, 3

direct summand, 3

irreducible, 2

semisimple, 3

simple, 2

monoid, 14

invertible element, 14

inverse element, 14

submonoid, 16

unit, 14

inverse element, 14

unit group, 14

monomial algebra, 23

Munn ring, 22

Perlis–Walker Theorem, 10

Rees matrix semigroup, 15

idempotents, 18

multiplication, 15

ring

idempotent, 2

nontrivial, 2

orthogonal, 2

primitive, 2

involution, 2

primitive central idempotents,

4, 12

simpe, 2

simple components, 4

ring of matrix type, 22

addition, 22

multiplication, 22

sandwich matrix, 22

semigroup, 14

0-simple, 19

completely 0-simple, 19

adjoining a zero, 15

adjoining an identity, 15

Brandt, 19

center, 17

congruence relation, 18

left, 18

Rees congruence, 18

right, 18

cyclic, 16

factor semigroup, 18

Rees factor semigroup, 18

homomorphism, 14

ideal, 17

generated by an element, 17

left, 17

right, 17

idempotent, 16

primitive, 16

identity, 14

inverse, 17

kernel, 17

Malcev nilpotent, 21

maximal subgroup, 17

null, 14

periodic, 16

periodic element, 16

Index

principal factor, 20

principal series, 20

factors, 20

regular, 17

Von Neumann condition, 17

relation J , 20

, 20

J(x), 20

J -class, 20

J -equivalence, 20

simple, 19

subsemigroup, 16

zero element, 14

semigroup algebra, 21

Isomorphism Problem, 40

semigroup ring, 21

coeﬃcient of s in α, 21

contracted, 22

Isomorphism Problem, 40

product of elements, 21

sum of elements, 21

support of an element, 21

We dderburn–Artin Theorem, 4

We dderburn–Malcev Theorem, 3

Index

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