Simple components in group algebras

Abstract

Let G be a finite group of exponent n, K a field of characteristic 0, and KG the group algebra of G over K. We write B(K) for the Brauer group of K, and C(K, G) for the collection of all simple summands of KG each of which has center K. If [A], E C(K, G) and {A} belongs to B(K), and we say that (A} is represented by C(K, G). This gives a natural multiplication to C(K, G), which may not be closed. This paper begins a study of C(K, G) with multiplication, to determine its closeness to a group structure. If G is an elementary group, we show that if {A ] is represented by C(K, G), then so is (A)’ for all r. If no restriction is put on G, then we can only show that r must be relatively prime to {A} in B(K). We shall write {A)‘= (A’}, where A r denotes the division algebra in the Brauer class (A )‘, and we say that b[d], E KG if