The Glauberman Character Correspondence

Abstract

[ES] Uno de los hechos fundamentales en la teoría de caracteres de grupos finitos es la correspondencia de Glauberman. Si un grupo A de orden potencia de un primo p actúa por automorfismos en un grupo finito G de orden no divisible por p, entonces G. Glauberman descubrió una biyección canónica entre el conjunto de caracteres complejos irreducibles fijados por A y los caracteres irreducibles del subgrupo de puntos fijos de A sobre G. Esta correspondencia está detrás de muchas las conjeturas en teoría de los caracteres.

[EN] One the most fundamental facts in the character theory of finite groups is the Glauberman correspondence: if a group A of order a power of a prime p acts by automorphisms on a finite group G of order not divisible by p, then G. Glauberman discovered a canonical bijection between the set of the irreducible complex characters of G fixed by A and the irreducible characters of the fixed point subgroup of A on G. This correspondence is behind many of the deep conjectures in character theory.

In our work, first we present a fairly elementary self-contained proof of the Glauberman correspondence, due to G. Navarro. In the second part, we show that the Glauberman correspondence commutes with irreducible induction and restriction of characters. This result, due to M. Isaacs and G. Navarro, is used in certain works on the Langlands program.

Keywords: character theory, Glauberman correspondence, group actions, character induction, character restriction.