A Note About Character Correspondence

Let S and G be finite groups such that S acts on G and let C= Cg{S). A main problem about character correspondence is that whether there exists a "natural" one-to-one correspondence * : Irrs(G)→ Irr(C), x→x~*Under the assumption of (|S|, |G|) = 1, we have two different situations. When S is solvable, in 1968, Glauberman gived the "natural" one-to-one correspondence which is called Glauberman correspondence. In the remaining case that S is unsolvable, since the Feit-Thompson theorem says that groups of odd order must be solvable, we know |S| is even and |G| is odd, that is to say, G is a solvable group of odd order. For this case, Isaacs proved that there also exists a canonical one-to-one map from Irrs(G) onto Irr(C) in 1973. This causes a natural question: how about the common situation? It is that S is solvable and G is a solvable group of odd order.This question was solved by Wolff1' in 1978. He proved that in the situation that both Glauberman correspondence and Isaacs correspondence are well defined, they are actually coincide. Obviously Wolf's work gives a useful technique for the study of the relationship between group actions and certain characters.Glauberman-Isaacs correspondence and some extentions of it are some of the hot topics in the character theory today. They have been studied deeply and many satisfactory results have been abtained in this direction. The main result of this thesis is an extention of the Glauberman-Isaacs correspondence in the weakly coprime situation. In fact, we get a canonical one-to-one correspondence in the case of (|S|, |G : N\) = 1 rather than (|S|, |G|) = 1. In particular, our Theorem 2.2 contains and generalizes many results that we have known.The following are main results of this thesis:Theorem 2.2 Let ( , G, N) be a coprime normal triple and H/N be a complement for G/N in T/N. Write C/N = CG/N(H). Then for each θ∈ Irr_H(N), there exists a uniquely defined bijection ρ : Irr_H(G|θ) → Irr(C|θ), x→ρ(X) Also ρ(x) is a constituent of xcCorollary 2.3 Let S and G be groups such that S act on G. Let N ≤ G is S-invariant. Assume that (|S|, |G : N|) = 1 and that G/N is solvable. Let θ G Irr_S(iV). Then θ~G has some S-invariant irreducible constituents.Corollary 2.4 Let (, G, N) be a coprime normal triple. Assume that one of /G or G/N is solvable. Let θ ∈ Irr(iV) be invariant in . Then θ~G has some -invariant irreducible consituents.