Finite Metabelian Group Algebras
Shalini Gupta
Department of Mathematics, Punjabi University, Patiala, India
email:[email protected]
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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algebras, J. Ramanujan Math Soc. 28(2) (2013) 141-158.
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