Character correspondences and subgroups of operator groups
Let \(A\) and \(G\) be finite groups of relatively prime orders and suppose that \(A\) acts on \(G\) as a group of automorphisms. Let \(C\) be the subgroup of \(G\) consisting of the elements fixed by all automorphisms of \(A\). The Glauberman-Isaacs correspondence \(\pi(G,A)\) is a one-to-one correspondence between the set \(\text{Irr}_A(G)\) of irreducible complex characters of \(G\) fixed by all elements of \(A\) and the set \(\text{Irr}(C)\) of all irreducible complex characters of \(C\). This correspondence has been studied in great detail, especially in the theory of solvable groups. Let \(B\) be a subgroup of \(A\) and let \(\chi\in\text{Irr}_A(G)\). It was once conjectured that \(\chi\pi(G,A)\) is an irreducible constituent of the restriction of \(\chi\pi(G,B)\) to \(C\), and T. R. Wolf proved that this is true if \(| G|\) is odd, if \(B\) is subnormal in \(A\), or if \(A\) and \(B\) have the same fixed point subgroup in \(G\) and \(G\) is solvable. Despite these positive results, the author of the present paper and \textit{T. R. Wolf} showed [Bull. Lond. Math. Soc. 29, No. 3, 307-308 (1997; Zbl 0866.20005)] that the conjecture is false in general, even if \(G\) is solvable. The main result of the paper under review is that the conjecture is true if either \(G\) is supersolvable or \(\chi\) is induced from an \(A\)-invariant irreducible character of an \(A\)-invariant supersolvable subgroup of \(G\). The final section of the paper examines related problems in the context of \(\pi\)-Brauer characters of \(\pi\)-separable groups, where \(\pi\) is a set of primes.
scientific article; zbMATH DE number 1213802