Wedderburn decomposition of finite group algebras

We show a method to effectively compute the Wedderburn decomposition and the primitive central idempotents of a semisimple finite group algebra of an abelian-by-supersolvable group G from certain pairs of subgroups of G. In this paper F = Fq denotes a finite field with q elements and G is a finite group of order n such that FG is semisimple, or equivalently (q, n) = 1. The group algebra FG is an algebraic object of major interest in pure and applied algebra. One of the remarkable applications of finite group algebras appears in coding theory because cyclic codes are ideals of group algebras of cyclic groups [10]. More generally, there is a long tradition on the study of abelian codes (ideals in finite abelian group algebras) or group codes (one sided ideals in arbitrary finite group algebras) [1, 2, 3, 5, 6, 11, 13, 14, 15]. One of the major motivations in the study of non cyclic group algebra codes relies on the fact that many important codes can be realized as ideals of a non cyclic group algebras [10, Chapter 9], [2, 15]. This paper focuses on the computation of the Wedderburn decomposition, that is the decomposition of FG as a direct sum of matrix rings over division rings. With this decomposition at hand it is straightforward to produce all the ideals of FG. If e1, . . . , em are the primitive central idempotents of FG then FG = FGe1 ⊕ . . .⊕ FGem is the Wedderburn decomposition of FG. However the idempotent themselves do not provide information on the structure of FGei’s as matrix rings of division rings. If G is cyclic then the primitive idempotents of FG are in one-to-one correspondence with the q-cyclotomic classes module |G|, the order of G [10] and using this it is not difficult to compute the primitive idempotents and the Wedderburn decomposition of any commutative finite group algebra (see Proposition 2). The primitive central idempotents of non commutative group algebras are more difficult to compute but can be obtained from the character table of the group. Recently Jespers, Leal and Paques [4] have introduced a character free method to compute the primitive central idempotents of a rational group algebra QG for G a finite nilpotent. This method has been extended and simplified in [8]. Further the results of [8] provides information on the structure of the simple components of QG, i.e. information on the Wedderburn decomposition of QG. The main aim of this paper is showing that this method can be used to compute the primitive central idempotents and the Wedderburn decomposition of FG. For example we show how to compute the Wedderburn decomposition and the primitive central idempotents of FG if G is abelian-by-supersolvable (see Theorem 7 and Corollary 8). We start establishing the basic notation. The algebraic closure of F is denoted by F̂. For every positive integer k coprime with q, ξk denotes a primitive k-th root of unity in F̂ and ok = ok(q) denotes the multiplicative order of q module k. Recall that F(ξk) = Fqok the field of order qok . If α ∈ FG and g ∈ G then αg = g−1αg and CenG(α) denotes the centralizer of α in G. The notation ∗Partially supported by the MECD of Spain, CAPES of Brazil and Fundación Séneca of Murcia