Correspondences and irreducible products of characters (Q1310415)

Suppose the finite group \(S\) acts on the finite group \(G\) with \((| G|, | S|) = 1\). Let \(\text{Irr}_ S (G)\) be the set of the irreducible characters of \(G\) fixed by \(S\); let \(\chi \in \text{Irr}_ S (G)\) and denote by \(\chi^* \in \text{Irr} (C_ G(S))\) the image of \(\chi\) under the Glauberman-Isaacs correspondence. The main theorems in this paper are the following. Theorem A. If \(\alpha \in \text{Irr}_ S (G)\), \(\beta \in \text{Irr}_ S (G)\) and \(\alpha \beta \in \text{Irr} (G)\), then \((\alpha \beta)^* = \alpha^* \beta^*\). Theorem B. Let \(\eta \in \text{Irr} (C_ G(T))\) be the \(T\)- correspondent of \(\chi \in \text{Irr}_ S (G)\), where \(T \leq S\). Suppose \(G\) is solvable. Then \(\chi^*(1)\) divides \(\eta (1)\), and also \(\eta(1) (\chi^*(1))^{-1}\) divides \(| C_ G (T) : C_ G (S)|\).