On character correspondences in \(\pi\)-separable groups (Q1813924)

Let \(A\) and \(G\) be finite groups and suppose \(A\) acts on \(G\) by automorphisms. We denote by \(\text{Irr}(G)\) the set of ordinary (complex) irreducible characters of \(G\). For a prime \(p\), \(\text{IBr}_ p(G)\) denotes the set of all irreducible Brauer characters of \(G\) with respect to \(p\). Let \(\pi\) be a set of prime numbers and let \(\pi'\) be the set of primes complementary to \(\pi\). For \(\chi \in \text{Irr}(G)\), we denote by \(\widehat{\chi}\) the restriction of \(\chi\) to the set \(\widehat{G}\) of all \(\pi\)-elements of \(G\). Now assume that \(A\) acts on \(G\) by automorphisms and \((| A|,| G|) = 1\). Under the assumption that \(A\) is solvable, Glauberman established a natural bijection from \(\text{Irr}(G)_ A\) onto \(\text{Irr}(C_ G(A))\). The purpose of this paper is to generalize this result to \(\pi\)-separable groups by applying Isaac's \(\pi\)-generalization of Brauer characters. Namely, we have the following theorem: Let \(A\) act on \(G\) such that \((| G|, | A|) = 1\). Suppose \(G\) is \(\pi\)-separable. Then there exists a natural bijection \(\widetilde{\Pi}(G,A): I_ \pi(G)_ A \to I_ \pi(C_ G(A))\) and the following hold. (1) If \(B \trianglelefteq A\), then \(\widetilde{\Pi}(G,A) = \widetilde{\Pi}(G,B) \widetilde{\Pi}(C_ G(B),A/B)\). (2) If \(A\) is a \(q\)-group for a prime \(q\) and \(\psi \in I_ \pi(G)_ A\), then \((\psi) \widetilde{\Pi}(G,A)\) is a unique \(\pi\)-irreducible constituent of \(\psi_{C_ G(A)}\) with multiplicity prime to \(q\).