Counting objects which behave like irreducible Brauer characters of finite groups

Abstract

Let N 0 G, where G is a finite group, so that G acts on Irr(N), the set of irreducible ordinary (complex) characters of N. Suppose 8 E Irr(N) is G-invariant and write Irr(G 10) to denote the set of those x E Irr(G) such that x,~ involves (and hence is a multiple of) 8. How large is Irr(G IO)? The answer’ to this question is well known, and not very hard to prove. Analogously to the fact that lIrr(G)I is equal to the number of conjugacy classes of G, it turns out that IIrr(GJB)l is equal to the number of certain “good” classes in the group G/N. There are two equivalent descriptions of which classes of G/N these are. One characterization depends on a certain complex twisted group algebra of G/N, and in this formulation, the result which counts Irr(G 1 t9) is due to I. Schur [S]. The other formulation of this result, due to P. X. Gallagher [Z], defines “goodness” internally to G; it depends on the character theory of G and its subgroups. Specifically, Gallagher defines an element of 2 E G/N to be O-good if some extension of 8 to (N, g) is invariant in C, where C/N= C, ,Jg). The property of being Q-good is preserved by conjugacy in GIN and Gallagher shows that lIrr(G I 13)l is equal to the number of classes of G:IN which consist of &good elements. Somewhat less well known is the corresponding result for irreducible Brauer characters with respect to some prime p. If 8 E IBr(N) is G-invariant, we intend to count the elements of IBr(G ( t9), i.e., the irreducible Brauer characters of G whose restrictions to N are multiples of 8. In view of the fact that IIBr(G)( equals the number of p-regular classes of G, it is reasonable to guess that IIBr(Gl f?)l is the number of e-good p-regular