GLAUBERMAN-ISAACS CORRESPONDENCE AND PI-BRAUER CHARACTERS

[摘要] Suppose that a group A acts on a group G of coprime order; then the Glauberman-Isaacs correspondence defines a bijection between the A-invariant irreducible characters of G and the irreducible characters of the fixed-point subgroup C = C-G(A). For a set of primes pi, and a pi-separable group G, the correspondent of an A-invariant pi-Brauer character phi was defined as follows: find chi is an element of B-pi(G) subset of or equal to Irr(G) which is the canonical lift for phi, then take the correspondent of chi, and finally restrict the correspondent of chi to the pi-elements of C. One of the main results of this paper is to show that the correspondent of an A-invariant pi-Brauer character is obtained if one chooses any of its A-invariant lifts in Irr(G) and applies the above algorithm. Thus, in the case where chi lifts a pi-Brauer character, we can say that application of the correspondent map commutes with application of the map which restricts characters to pi-elements of the group. We show by example that these maps do not commute when chi is the lift of a sum of A-invariant pi-Brauer characters, and prove a theorem characterizing this behavior when A is solvable. (C) 1994 Academic Press, inc.

[发布日期] 1994-11-01