Characters, coprime actions, and operator groups
The following result is proved. Main Theorem. Let \(M\) be a normal subgroup of a finite group \(G\). Assume that \(M\) has a normal Hall subgroup \(N\) of odd order and that \(S\) is a complement for \(N\) in \(M\). If \(\chi\in\text{Irr}(N)\) is a \(G\)-invariant character, then \(\chi\) extends to \(G\) if and only if \(\chi^*\) extends to \(N_G(S)\), where \(\chi^*\in\text{Irr}(C_M(S))\) is the Glauberman-Isaacs correspondent of \(\chi\). Corollary. Let \(S\) act on \(G\) so that \((| S|,| G|)=1\) and that \(| G|\) is odd. Assume that \(A\) acts on \(SG\) so that \(S\) and \(G\) are both \(A\)-invariant under its action. If the \(S\)-invariant character \(\chi\) is \(A\)-invariant, then \(\chi\) extends to \(AG\) if and only if \(\chi^*\) extends to \(AC_G(S)\), where \(\chi^*\in\text{Irr}(C_G(S))\) is the Glauberman-Isaacs correspondent of \(\chi\).
scientific article; zbMATH DE number 1145411