Vol. 20, No. 1, 1967

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Representations of direct products of finite groups

Burton I. Fein

Vol. 20 (1967), No. 1, 45–58

Let G be a finite group and K an arbitrary field. We denote by K(G) the group algebra of G over K. Let G be the direct product of finite groups G1 and G2,G = G1 ×G2, and let Mi be an irreducible K(Gi)-module, i = 1,2. In this paper we study the structure of M1,M2, the outer tensor product of M1 and M2.

While M1,M2 is not necessarily an irreducible K(G)-module, we prove below that it is completely reducible and give criteria for it to be irreducible. These results are applied to the question of whether the tensor product of division algebras of a type arising from group representation theory is a division algebra.

We call a division algebra D over K K-derivable if DHomK(G)(M,M) for some finite group G and irreducible K(G)-module M. If B(K) is the Brauer group of K, the set B0(K) of classes of central simple K-algebras having division algebra components which are K-derivable forms a subgroup of B(K). We show also that B0(K) has infinite index in B(K) if K is an algebraic number field which is not an abelian extension of the rationals.

Mathematical Subject Classification
Primary: 20.80
Secondary: 16.40
Received: 7 October 1965
Published: 1 January 1967
Burton I. Fein