Finite Metabelian Group Algebras

Shalini Gupta

Department of Mathematics, Punjabi University, Patiala, India

email:gupta_math@yahoo.com

Keywords: semisimple group algebra, primitive central idempotents, Wedderburn decomposition, metabelian groups.

Abstract. Given a finite metabelian group G, whose central quotient is abelian (not cyclic) group of

order 2p , p odd prime, the objective of this paper is to obtain a complete algebraic structure of

semisimple group algebra 𝔽𝑞[𝐺] in terms of primitive central idempotents, Wedderburn

decomposition and the automorphism group.

1. Introduction

Let F be a field and G be a finite group such that the group algebra ][GF is semisimple.

A fundamental problem in the theory of group algebras is to understand the complete algebraic

structure of semisimple group algebra ][GF . In the recent years, a lot of work has been done to

solve this problem [1,2,5,7,8,9]. Bakshi et.al [3] have solved this problem for semisimple finite

metabelian group algebras 𝔽𝑞[𝐺], where 𝔽𝑞 is a finite of order q and G is a finite metabelian

group. They further illustrated their algorithm by explicitly finding a complete set of primitive

central idempotents, Wedderburn decomposition and the automorphism group of semisimple group

algebra of certain groups whose central quotient is Klein’s four-group. In the present paper, a

complete algebraic structure of semisimple group algebra 𝔽𝑞[𝐺] for some finite groups G, whose

central quotient , )(GZG , is the direct product of two cyclic groups of order p , p odd prime, is

obtained. It is known [6] that finitely generated groups G , whose central quotient is isomorphic to

ℤ𝑝 × ℤ𝑝 break into nine classes. The complete algebraic structure of 𝔽𝑞[𝐺], for group G in the

two of the nine classes, is obtained in the present paper .

2. Notation

Let G be a finite group of order coprime to q and )(GIrr denotes the set of all irreducible

characters of G over�̅�𝑞, the algebraic closure of 𝔽𝑞. Let GKH such that HK is cyclic of

order n and ),()( KNHNT GG where )(HNG denotes the normalizer of H in G . Let

𝒞(𝐾 𝐻⁄ )denotes the set q -cyclotomic sets of )( HKIrr containing the generators of )( HKIrr .

Suppose that T act on 𝒞(𝐾 𝐻⁄ ) by conjugation, then it is easy to see that stabilizer of any

𝐶 ∈ 𝒞(𝐾 𝐻)⁄ remains the same. Let )( HKEG denotes the stabilizer of any𝐶 ∈ 𝒞(𝐾 𝐻)⁄ and let

)( HK denotes the set of distinct orbits of𝒞(𝐾 𝐻⁄ )under the action of T on𝒞(𝐾 𝐻⁄ ). Observe

that

|)(| HK = ,)(||

|)(|)(

qordT

HKEn

n

G

where )(qord n denotes the order of q modulo n .

International Journal of Pure Mathematical Sciences Submitted: 2016-05-30ISSN: 2297-6205, Vol. 17, pp 30-38 Revised: 2016-08-06doi:10.18052/www.scipress.com/IJPMS.17.30 Accepted: 2016-08-192016 SciPress Ltd, Switzerland Online: 2016-10-24

SciPress applies the CC-BY 4.0 license to works we publish: https://creativecommons.org/licenses/by/4.0/

For 𝐶 ∈ 𝒞(𝐾 𝐻)⁄ , 𝜒 ∈ 𝐶 and n a primitive nth root of unity in �̅�𝑞, set

)( HKC = ,)))(((|| 1

/)(

1

Kg

FF ggtrKqnq

and ),,( HKGeC as the sum of distinct G -conjugates of )( HKC .

3. Metabelian group algebras

The notation used in [4] will be followed: For a normal subgroup N of G , let NAN be an

abelian normal subgroup of NG of maximal order. Let ND be the set of subgroups ND of

NAN such that DAN is cyclic and NGT be the set of representatives of ND under the

equivalence relation of conjugacy of subgroups of NG . Define

NDTNDNANDS NGNNG ,|),{(: core-free in }NG .

Let

}.),(,,|),,{(: NGNNGN SNANDSGNNANDNS

We are now ready to recall the theorem describing the complete algebraic structure of

semisimple finite metabelian group algebras:

Theorem 1 [3]: Let G be a finite metabelian group of order coprime to q . Then,

(i) A complete set of primitive central idempotents of semisimple group algebra 𝔽𝑞[𝐺] is given by

the set )};(,),,(|),,({ DACSNANDNDAGe NNNC

(ii) the simple component corresponding to primitive central idempotent ),,( DAGe NC is

),(),,(][ ),(]:[ DNAoN qAGNCq FMDAGeGF

where )(RM n denotes the ring of 𝑛 × 𝑛 matrices over the ring R and

]:),([

)(),(

]):([

NNG

DA

NADAE

qordDAo N . Moreover the number of such simple components is |),(| DAN .

4. Groups whose central quotient is abelian (not cyclic) group of order 2p

Conelissen and Milies [6] have classified indecomposable finitely generated groups G , such

that pp CCGZG )( , into nine classes. In all of these classes, )(,, GZbaG with some more

relations as described in following table:

International Journal of Pure Mathematical Sciences Vol. 17 31

Group G )(GZ Relations

𝔊1 c ,,,, )(11 1 mm pppp bacabcba

1m

𝔊2 c ,,,, )(1111 1 mm pppp bacabccbca

1m

𝔊3 21 cc ,,,,,)(

1

11

21

1

2

1121

mmmppppp bacabcccba

1, 21 mm

𝔊4 21 cc ,,,,,)(

1

11

21

1

2

1

1

1121

mmmppppp bacabcccbca

1, 21 mm

𝔊5 11 uc ,,,,)(

1

11

1

1

1

111

mmpppp bacabcuba

11 m

𝔊6 11 uc ,,,,)(

1

11

1

1

1

1

1

111

mmpppp bacabcubca

11 m

𝔊7 321 ccc ,,,,,,)(

1

11

321

1

3

1

2

11321

mmmmpppppp bacabccccbca

1,, 321 mmm

𝔊8 121 ucc ,,,,,)(

1

11

21

1

1

1

2

1121

mmmppppp bacabccubca

1, 21 mm

𝔊9 211 uuc ,,,,)(

1

11

1

1

2

1

1

111

mmpppp bacabcubua

11 m

2,1),( iuo i is infinite.

It can be see easily that G is finite metabelian group only in five classes. Out of these five

classes, we will give a complete algebraic structure of𝔽𝑞[𝐺], for 𝐺 = 𝔊1 and 𝔊2only. The rest of

the cases can be dealt similarly. Throughout this section 𝔽𝑞 is a finite field with q elements and

1),gcd( qp . Let )(qord p , the order of q modulo p , be f and f

pe

1 . Write

,1 cpq df where p does not divide c . Then for ,1l

.1,

,,)(1

dlfp

dlfqord

dlp

4.1. Structure of 𝔽𝑞[𝔊1]

Let G be a group of type 𝔊1. Thus G has following representation:

𝐺 = ⟨𝑎, 𝑏, 𝑐 | 𝑎𝑝 = 𝑏𝑝 = 𝑐𝑝𝑚= 1, 𝑏−1𝑎−1𝑏𝑎 = 𝑐𝑝

𝑚−1, 𝑐 𝑐𝑒𝑛𝑡𝑟𝑎𝑙 𝑖𝑛 𝐺⟩(1)

32 IJPMS Volume 17

where p is prime and 1m . For 2p , the complete algebraic structure of 𝔽𝑞[𝐺] can be read

from [3]. Suppose p is an odd prime. For 2m , define

,10,,,:,,:,,:,1:)(

3

)(

210 picbaKbacKacKKipiii

,,:,,:,,:),,(

6

),(

5

),(

4 kipjipkjijipjijipjibcacKacbKbcaK

1,1,20 pkjmi .

Theorem 2. Acomplete set of primitive central idempotents of semisimple group algebra 𝔽𝑞[𝐺],

G of type 𝔊1, is given as follows:

Primitive central idempotents of 𝔽𝑞[𝐺] for 1m :

).,(),,,,(

);,(),,,,(

;10),,(),,,,(

);(),,,(

acaCacaGe

caGCcaGGe

pibacGCbacGGe

GGCGGGe

C

C

ii

C

C

Primitive central idempotents of 𝔽𝑞[𝐺] for 2m :

);(),,,(

);(),,,(

11

11

KGCKGGe

aKCaKGe

C

C

.1,1,20)(),,,(

;11,20)(),,,(

;11,20)(),,,(

;10)(),,,(

;10)(),,,(

),,(

6

),,(

6

),(

5

),(

5

),(

4

),(

4

)(

3

)(

3

)(

2

)(

2

pkjmiKGCKGGe

pjmiKGCKGGe

pjmiKGCKGGe

piKGCKGGe

piKGCKGGe

kjikji

C

jiji

C

jiji

C

ii

C

ii

C

To prove this Theorem, we first need to find the normal subgroups of G .

Lemma 1. LetG be a group as defined in (1) and 𝒩 be the set of distinct normal subgroups of

G . Then

(i) For m=1, 𝒩 = {⟨1⟩, ⟨𝑐⟩, ⟨𝑐, 𝑎⟩, ⟨𝑐, 𝑎, 𝑏⟩⟨𝑐, 𝑎𝑖𝑏⟩, 0 ≤ 𝑖 ≤ 𝑝 − 1}

and )}1,1,{()},,1,,{()},,,1{( GacGaccaaS

)};10|,,1,,{( pibacGbac ii

(ii) For 𝑚 ≥ 2, 𝒩 = {⟨𝑐𝑝𝑖⟩ , ⟨𝑐𝑝

𝑖, 𝑎⟩ , ⟨𝑐𝑝

𝑖, 𝑏⟩ , ⟨𝑐𝑝

𝑖, 𝑎, 𝑏⟩ | 0 ≤ 𝑖 ≤ 𝑚 − 1} ∪

,20|,,,,,,,,,{1

miacbbcabacbcacbac jipjipjipjipjipjmp

}1,1,20|,,{}11 pkjmibcacbacpj kipjipjkjip

and }10|),,{()},,{()},,{()(

20

)(

210110 piKGKKKGKKKaKSii

jmiKGKKpiKGKKjijiii

1,20|),,{(}10|),,{(),(

40

),(

4

)(

30

)(

3

|),,{(}11,20|),,{(}1),,(

60

),,(

6

),(

50

),(

5

kjikjijijiKGKKpjmiKGKKp

}.1,1,20 pkjmi

International Journal of Pure Mathematical Sciences Vol. 17 33

Proof. It can be seen easily that in (i) and (ii), the subgroups listed are distinct and normal in G .

Also if ,GN then it can be shown easily, as in [[3], Lemma 4], that N is one of the subgroups

listed in the statement of Lemma.

Observe that in both (i) and (ii), for acNAN N ,,1 . Hence )},,{( acaS NG.

Moreover for non-identity normal subgroup N of G , the derived group of G , 1

'mpcG is

contained in N , thus NG is abelian and hence NGNAN . Thus for all non identity normal

subgroups N of G ,

.,

,)},,1{(

otherwise

cyclicisNGifNGS NG

Thus to complete the proof, we need to find only those𝑁 ∈ 𝒩 for which NG , is cyclic. In

(i), the subgroups 10,,,,,,, pibacbacac i have cyclic quotient with G , whereas in (ii),

the following normal subgroups have cyclic quotient with G :

,10,,,)(

3

)(

21 piKKKii

1,1,20,,,),,(

6

),(

5

),(

4 pkjmiKKKkjijiji

.

Thus the proof of the lemma is complete.

Proof of Theorem 2. The list of primitive central idempotents of group algebra 𝔽𝑞[𝐺] can now be

easily obtained with the help of Theorem 1 and Lemma 1.

Theorem 3. TheWedderburn decomposition and the automorphism group of semisimple group

algebra 𝔽𝑞[𝐺], G of type 𝔊1, are given as follows:

Wedderburn decomposition

𝔽𝑞[𝐺] ≅

{

𝔽𝑞⨁𝔽𝑞𝑓

((𝑝+1)𝑒)⨁𝑀𝑝 (𝔽𝑞𝑓)(𝑒), 𝑚 = 1,

𝔽𝑞⨁ 𝔽𝑞𝑓(𝑝𝑚+1−1

𝑓)⨁ 𝑀𝑝 (𝔽𝑞𝑓)

(𝑝𝑚−1𝑒), 2 ≤ 𝑚 ≤ 𝑑,

𝔽𝑞⨁𝔽𝑞𝑓(𝑝𝑑+2−1

𝑓)⨁ ∑ 𝔽

𝑞𝑓𝑝𝑖−𝑑

(𝑝𝑑+1𝑒)

𝑚−1

𝑖=𝑑+1

⨁𝑀𝑝 (𝔽𝑞𝑓𝑝𝑚−𝑑)(𝑝𝑑−1𝑒)

, 𝑚 ≥ 𝑑+ 1.

Automorphism group

𝐴𝑢𝑡(𝔽𝑞[𝐺]) ≅

{

(ℤ𝑓

((𝑝+1)𝑒) ⋉ 𝑆(𝑝+1)𝑒) ⊕ ((𝑆𝐿𝑝(𝔽𝑞𝑓) ⋉ ℤ𝑓)(𝑒) ⋉ 𝑆𝑒),𝑚 = 1,

(ℤ𝑓(𝑝𝑚+1−1

𝑓)⋉ 𝑆

(𝑝𝑚+1−1

𝑓)) ⊕

((𝑆𝐿𝑝(𝔽𝑞𝑓) ⋉ ℤ𝑓)(𝑝𝑚−1𝑒) ⋉ 𝑆𝑝𝑚−1𝑒), 2 ≤ 𝑚 ≤ 𝑑,

(ℤ𝑓(𝑝𝑑+2−1

𝑓)⋉ 𝑆

(𝑝𝑑+2−1

𝑓)) ⊕ ∑ (ℤ𝑓𝑝𝑖−𝑑

(𝑝𝑑+1)𝑒) ⋉ 𝑆𝑝𝑑+1𝑒)⨁

𝑚−1

𝑖=𝑑+1

((𝑆𝐿𝑝(𝔽𝑞𝑓𝑝𝑚−𝑑) ⋉ ℤ𝑓𝑝𝑚−𝑑)

(𝑝𝑑−1𝑒) ⋉ 𝑆𝑝𝑑−1𝑒) ,𝑚 ≥ 𝑑 + 1.

whereℤ𝑛 denotes the cyclic group of order n , nS denotes the symmetric group of degree n and

for a group H , )(nH a direct sum of n copies of H .

34 IJPMS Volume 17

Proof of Theorem 3. In order to find the Wedderburn decomposition of 𝔽𝑞[𝐺], we need to find the

simple component corresponding to each primitive central idempotent. More precisely, for each

,)(,),,( DACSNANDN NN we need to calculate ),( DAo N and |)(| DAN , as given by

the following tables:

Case I : 1m

),,( NANDN N )( DAE NG ),( DAo N |)(| DAN

),,,1( caa ca, f e

),,1,,( acGac G f e

)1,1,( G G 1 1

),,1,,( bacGbac ii

10 pi

G f e

Case II : 2m

),,( NANDN N )( DAE NG ),( DAo N |)(| DAN

),,( 10 KaK 1K

.1,

,,

dmfp

dmf

dm

.1,

,,

1

1

dmep

dmep

d

m

),,( 101 KGKK G f e

),,()(

20

)(

2

iiKGKK

10 pi

G f e

),,()(

30

)(

3

iiKGKK

10 pi

G

1,

1,

0,1

difp

dif

i

di

.1,

,1,

,0,1

1

1

diep

diep

i

d

i

),,(),(

40

),(

4

jijiKGKK

11

,20

pj

mi

G

.,

,1,

1 difp

dif

di

.,

,1,

1 diep

diep

d

i

),,(),(

50

),(

5

jijiKGKK

11

,20

pj

mi

G

.,

,1,

1 difp

dif

di

.,

,1,

1 diep

diep

d

i

),,(),,(

60

),,(

6

kjikjiKGKK

1,1

,20

pkj

mi

G

.,

,1,

1 difp

dif

di

.,

,1,

1 diep

diep

d

i

Now, the required Wedderburn decomposition and automorphism group can be easily read

from these two tables and [3, Theorem 3].

International Journal of Pure Mathematical Sciences Vol. 17 35

4.2. Structure of 𝔽𝑞[𝔊2]

Observe that if group G is of type 𝔊2, then it has the following presentation:

𝐺 = ⟨𝑎, 𝑏| 𝑎𝑝𝑚+1

= 1, 𝑏𝑝 = 𝑎𝑝, 𝑏−1𝑎−1𝑏𝑎 = 𝑎𝑝𝑚+1, 𝑎𝑝 𝑐𝑒𝑛𝑡𝑟𝑎𝑙 𝑖𝑛 𝐺⟩,

where p is a prime and 1m . For 2p , the complete algebraic structure of 𝔽𝑞[𝐺] can be read

from [3]. Suppose p is an odd prime. For 1m , set:

,,:,1,:,:,1: 3

)(

210 baLmiaLaLLipi

,10,,:)(

4 pibaaL ipi

.1,2,,: 1),(

5

1

pjmibaaLii jppji

The following Theorems give a complete algebraic structure of semisimple group algebra

𝔽𝑞[𝐺]:

Theorem 4. A complete set of primitive central idempotents of semisimple group algebra 𝔽𝑞[𝐺],

G of type 𝔊2, is given as follows:

Primitive central idempotents of 𝔽𝑞[𝐺]

𝑒𝐶(𝐺,𝐺, 𝐺), 𝐶 ∈ ℜ(𝐺 𝐺⁄ ); 𝑒𝐶(𝐺,𝐺, 𝐿1), 𝐶 ∈ ℜ(𝐺 𝐿1⁄ ); 𝑒𝐶(𝐺,𝐺, 𝐿3), 𝐶 ∈ ℜ(𝐺 𝐿3⁄ ); 𝑒𝐶(𝐺, 𝐿1, 𝐿0), 𝐶 ∈ ℜ(𝐿1 𝐿0⁄ );

𝑒𝐶 (𝐺,𝐺, 𝐿4(𝑖)) , 𝐶 ∈ ℜ (𝐺 𝐿4

(𝑖)⁄ ) , 0 ≤ 𝑖 ≤ 𝑝 − 1;

𝑒𝐶 (𝐺,𝐺, 𝐿5(𝑖,𝑗)) , 𝐶 ∈ ℜ (𝐺 𝐿5

(𝑖,𝑗)⁄ ) , 2 ≤ 𝑖 ≤ 𝑚, 1 ≤ 𝑗 ≤ 𝑝.

Proof of Theorem 4. In view of Theorem 1, to find a complete list of primitive central idempotents

of 𝔽𝑞[𝐺] we first need to list all normal subgroups of G . It can be see easily that the set 𝒩 of

distinct normal subgroups of G is equal to

}.1,2,,10,,,1,,,{),(

5

)(

43

)(

210 pjmiLpiLLmiLLLjiii

For 10 , LNALN N . Hence )},{( 10 LLS NG . Moreover, for non-identity𝑁 ∈ 𝒩,NGS

is non-empty if and only if NG is cyclic. The following 𝑁 ∈ 𝒩 have cyclic quotient with G :

.1,2,,10,,,),(

5

)(

431 pjmiLpiLLLjii

Thus (i) follows from Theorem 1.

Theorem 5. The Wedderburn decomposition and the automorphism group of semisimple group

algebra 𝔽𝑞[𝐺], G of type 𝔊2, are as follows:

Wedderburn decomposition

𝔽𝑞[𝐺] ≅

{

𝔽𝑞⨁ 𝔽𝑞𝑓

(𝑝𝑚+1−1

𝑓)⨁𝑀𝑝 (𝔽𝑞𝑓)

(𝑝𝑚−1𝑒), 𝑚 ≤ 𝑑− 1,

𝔽𝑞⨁𝔽𝑞𝑓(𝑝𝑑+1−1

𝑓)⨁ ∑ 𝔽

𝑞𝑓𝑝𝑖−𝑑

(𝑝𝑑𝑒)

𝑚

𝑖=𝑑+1

⨁𝑀𝑝 (𝔽𝑞𝑓𝑝𝑚−𝑑)(𝑝𝑑−1𝑒)

, 𝑚 ≥ 𝑑.

36 IJPMS Volume 17

Automorphism group

Aut(𝔽𝑞[𝐺])≅

{

(ℤ𝑓

(𝑝𝑚+1−1

𝑓)⋉ 𝑆

(𝑝𝑚+1−1

𝑓)) ⊕

(𝑆𝐿𝑝(𝔽𝑞𝑓) ⋉ ℤ𝑓)(𝑝𝑚−1𝑒) ⋉ 𝑆𝑝𝑚−1𝑒), 𝑚 ≤ 𝑑 − 1

(ℤ𝑓(𝑝𝑑+1−1

𝑓)⋉ 𝑆

(𝑝𝑑+1−1

𝑓)) ⊕ ∑ (ℤ𝑓𝑝𝑖−𝑑

(𝑝𝑑)𝑒) ⋉ 𝑆𝑝𝑑𝑒)⨁𝑚𝑖=𝑑+1

((𝑆𝐿𝑝(𝔽𝑞𝑓𝑝𝑚−𝑑) ⋉ ℤ𝑓𝑝𝑚−𝑑)

(𝑝𝑑−1𝑒) ⋉ 𝑆𝑝𝑑−1𝑒) ,𝑚 ≥ 𝑑.

Proof of Theorem 5. We will first find )( 01 LLEG .Observe that 1

01 || mpLL and

GLLEL G )( 011 . Let 1 dm . In this case, )( 01 LLEb G , if and only if iqmp

1 for

some fii 1, , where is a primitive 1mp th root of unity. This implies that

)(mod1 1 mim pqp , i.e.,),gcd( fi

fp , which gives that p divides 1p , a contradiction.

Hence in this case 101 )( LLLEG . For GLLEdm G )(, 01 . Thus we have the following:

),,( NANDN N )( DAE NG ),( DAo N |)(| DAN

),,( 00 LLG G 1 1

),,( 101 LGLL G f e

),,,()(

40

)(

4

iiLGLL

10 pi

G f e

),,,(),(

50

),(

5

jijiLGLL

pjmi 1,2

G

.1,

,,

difp

dif

di

.1,

,,

1

1

diep

diep

d

i

),,( 100 LLL

.,

,1,1

dmG

dmL

.,

,1,

dmfp

dmf

dm

.,

,1,

1

1

dmep

dmep

d

m

The Wedderburn decomposition and automorphism group of 𝔽𝑞[𝐺] can now be easily read

with the help of this table and [3, Theorem 3].

Acknowledgment

The author is grateful to the referees for their valuable suggestions which have helped to write

the paper in the present form.

International Journal of Pure Mathematical Sciences Vol. 17 37

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[2] G.K. Bakshi, S. Gupta, I.B.S. Passi, The structure of finite Semisimple metacyclic group

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