Induction of characters, kernels and local subgroups
It was proved by \textit{G. R. Robinson} [J. Algebra 114, No. 1, 53-57 (1988; Zbl 0656.20004)] that \((\alpha_{N_K(P)})^{N_G(P)}\in\text{Irr}(N_G(P))\), where \(P\) is a Sylow \(p\)-subgroup of \(K\), \(\alpha\in\text{Char}(K)\), \(\alpha^G\in\text{Irr}(G)\), \(\alpha(1)\leq p/2\). In this paper a generalization is shown in the sense that the same result follows from the condition ``\(P\text{Ker}(\alpha)\vartriangleleft K\)'', but with ``\(\alpha(1)\leq p/2\)'' dropped. As a corollary it follows that if \(K/\ker(\alpha)\) is a \(p'\)-group, \((\alpha_{N_K(P)})^{N_G(P)}\) is a \(p\)-defect zero character of \(N_G(P)/P\). In order to achieve all this, three elementary lemmas with proofs are stated. The following is also shown in this paper; its proof turned out to be somewhat hard. Theorem. Suppose \(\chi=\alpha^G\in\text{Irr}(G)\), where \(K\leq G\), \(\alpha\in\text{Irr}(K)\) and \(K/\ker(\alpha)\) is a \(p'\)-group. Suppose \(G\) has a normal \(p\)-complement \(M\). Then \(MK\) is the stabilizer in \(G\) of the character \((\alpha^{MK})_M\in\text{Irr}(M)\).
scientific article; zbMATH DE number 1744068