The modular group algebra of certain finite groups with normal sylow p-subgroups

Abstract

The modular group algebras we consider are of the form kP >a Q where P is a finite p-group, Q a finite p’ group, k a splitting field of characteristic p for Q, and the action of Q on P arises from a homomorphism a: Q + Aut P with abelian image A. (By the Schur-Zassenhaus theorem (cf. [3]), every finite group G with a normal Sylow p-subgroup P a G is a semidirect product P ~1 G/P; but of course the action need not be abelian.) Theorem 1.1 is a special case of Theorem 1.3, stated below, which gives sufficient conditions for an isomorphism of group algebras of the form kP >a Q. Theorem 1.3 in turn is an application of Theorem 1.2 which gives an explicit description of the blocks of such a group algebra. We note that we are dealing here with a class of blocks which do not in general have finite representation type. and indeed can have any p-group as defect group (the defect group is P). Theorem 1.2 can be regarded as lifting the Wedderburn theorem for kQ up to kP >a Q in accordance with the numerical data about blocks provided by Brauer’s theorem (cf. [ 1, p. 6341). It expresses kP ~1 Q as a direct sum of full