Finite Metabelian Group Algebras

Given a finite metabelian group G, whose central quotient is abelian (not cyclic) group of order 2 p , 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.


Let
F be a field and G be a finite group such that the group algebra ] [G F is semisimple. A fundamental problem in the theory of group algebras is to understand the complete algebraic structure of semisimple group algebra ] [G F . 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 [ ], where 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 [ ] for some finite groups G, whose central quotient , , 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 ℤ × ℤ break into nine classes. The complete algebraic structure of [ ], for group G in the two of the nine classes, is obtained in the present paper .

Notation
Let G be a finite group of order coprime to q and ) (G Irr denotes the set of all irreducible characters of G over ̅ , the algebraic closure of . For ∈ ( ) ⁄ , ∈ and n  a primitive nth root of unity in ̅ , set

Metabelian group algebras
The notation used in [4] will be followed: For a normal subgroup N of G , let 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 [ ] is given by the set )}; . Moreover the number of such simple components is

Groups whose central quotient is abelian (not cyclic) group of order
2 p Conelissen and Milies [6] have classified indecomposable finitely generated groups G , such that , into nine classes. In all of these classes, with some more relations as described in following table: International Journal of Pure Mathematical Sciences Vol. 17 31 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 of [ ], for = 1 and 2 only. The rest of the cases can be dealt similarly. Throughout this section is a finite field with q elements and 1 ) , Let G be a group of type 1 . Thus G has following representation: G of type 1 , is given as follows: , , To prove this Theorem, we first need to find the normal subgroups of G .

Lemma 1.
Let G be a group as defined in (1) and be the set of distinct normal subgroups of , Proof. It can be seen easily that in (i) and (ii), the subgroups listed are distinct and normal in G . Also if , G N  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 . Thus for all non identity normal subgroups N of G , Thus to complete the proof, we need to find only those ∈ for which N G , is cyclic. In (i), the subgroups have cyclic quotient with G , whereas in (ii), the following normal subgroups have cyclic quotient with G : Thus the proof of the lemma is complete.

Proof of Theorem 2. The list of primitive central idempotents of group algebra
[ ] can now be easily obtained with the help of Theorem 1 and Lemma 1.

IJPMS Volume 17
Proof of Theorem 3. In order to find the Wedderburn decomposition of [ ], we need to find the simple component corresponding to each primitive central idempotent. More precisely, for each , , as given by the following tables: Now, the required Wedderburn decomposition and automorphism group can be easily read from these two tables and [3, Theorem 3].

Structure of
Observe that if group G is of type 2 , then it has the following presentation: The following Theorems give a complete algebraic structure of semisimple group algebra [ ]: . Moreover, for non-identity ∈ , N G S is non-empty if and only if N G is cyclic. The following ∈ have cyclic quotient with G : Thus (i) follows from Theorem 1. Proof of Theorem 5. We will first find ) (