Extending irreducible representations of normal subgroups
Let H be a normal subgroup of a finite group G, F a field, and V an irreducible FH module. The author wishes to find necessary and sufficient conditions in order that V can be extended to an FG module. Let \(D=End_{FH}(V)^ 0\) be the opposite endomorphism ring of V and let Z denote the multiplicative group of the center of D. If the Schur index m(V) of V is relatively prime to the order of G/H, then the author associates to V a cohomology class \(w_ G(V)\in H^ 2(G/H,Z)\). The main theorem then states that V can be extended to an FG module if and only if \(w_ G(V)\) vanishes. The proof uses a Hochschild-Serre exact sequence and a ''Schur-Zassenhaus type'' splitting theorem. Several interesting facts are presented in the course of proving the theorem. If H is a Hall subgroup, this extension problem, without the assumption on the Schur index, had been considered by \textit{I. Isaacs} [J. Algebra 68, 54-74 (1981; Zbl 0449.20016)] and by \textit{E. Dade} [ibid. 72, 374-403 (1981; Zbl 0497.20001)].
scientific article; zbMATH DE number 3991752
scientific article