Glauberman-Isaacs Correspondence and Π-Brauer Characters

AbstractSuppose 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 = CG(A). For a set of primes π, and a π-separable group G, the correspondent of an A-invariant π-Brauer character φ was defined as follows: find χ ∈ Bπ(G) ⊆ Irr(G) which is the canonical lift for φ, then take the correspondent of χ, and finally restrict the correspondent of χ to the π-elements of C. One of the main results of this paper is to show that the correspondent of an A-invariant π-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 χ lifts a π-Brauer character, we can say that application of the correspondent map commutes with application of the map which restricts characters to π-elements of the group. We show by example that these maps do not commute when χ is the lift of a sum of A-invariant π-Brauer characters, and prove a theorem characterizing this behavior when A is solvable