On normal subgroups and simple modules with a given vertex in a \(p\)-solvable group.
Let \(F\) be an algebraically closed field of characteristic \(p>0\), and let \(Q\) be a \(p\)-subgroup of a finite \(p\)-solvable group \(G\). For a normal subgroup \(N\) of \(G\) and a simple \(FN\)-module \(M\), the set of isomorphism classes of simple \(FG\)-modules with vertex \(Q\) lying over \(M\) is denoted by \(\text{IRR}^Q(FG|M)\). The author proves that \(|\text{IRR}^Q(FG|M)|=|\bigcup_{x\in G}\text{IRR}^Q(FNN_G(Q)|M^*)|\). This generalizes results of the author [Arch. Math. 84, No. 5, 385-391 (2005; Zbl 1074.20005)], of \textit{T. Okuyama} (unpublished) and of \textit{I. M. Isaacs} and \textit{G. Navarro} [J. Algebra 177, No. 2, 339-366 (1995; Zbl 0843.20008)]. The author's result is deduced from a more general theorem where the prime \(p\) is replaced by \(\pi\), a set of primes.
scientific article; zbMATH DE number 5144420