Actions and invariant character degrees. III
This paper is a sequel to two papers of the second author [published in J. Algebra 160, No. 1, 172-178 (1993; Zbl 0805.20007) and ibid. 184, No. 2, 397-406 (1996; Zbl 0868.20004)]. Among others, the following is shown. Theorem A: Suppose the finite group \(A\) acts coprimely on a finite \(\pi\)-separable group \(G\); let \(H\) and \(K\) be an \(A\)-invariant Hall \(\pi\)-subgroup and a \(\pi\)-complement of \(G\), respectively. Then every \(A\)-invariant irreducible character of \(H\) extends to \(G\) if and only if \(C_G(A)\) normalizes \(K\). Theorem D (Same assumptions as before): Every \(A\)-invariant irreducible character of \(G\) has \(\pi\)-degree if and only if \(C_G(A)\) normalizes \(K\) and \(C_K(A)=1\).
scientific article; zbMATH DE number 1019688