Fusion of \(\pi\)-subgroups, isometries, and normal complements (Q1821864)

The stated aim of the paper is ''to isolate some group-theoretic requirements which will ensure that a subgroup, H, of the finite group G will control the conjugacy of certain of its elements, in such a manner that generalized characters of H will extend to generalized characters of G.'' The type of result obtained is exemplified by Lemma 2: Let H be a subgroup of group G such that H controls elementary \(\pi\)-fusion for some set of primes, \(\pi\). Then (i) any generalized character of H which is constant on \(\pi\)-sections may be extended in a unique fashion to a generalized character of G which is constant on \(\pi\)-sections, and (ii) \(O^{\pi}(G)\cap H=O^{\pi}(H)\). (Note: A subgroup H of finite group G is said to control elementary \(\pi\)-fusion in G if, for every Brauer elementary \(\pi\)-subgroup E of H, \(N_ G(E)=C_ G(E)N_ H(E)\), and, for every \(\pi\)-element x of H, \([C_ G(x):C_ H(x)]\) is a (\(\pi\) '\(\cup \pi (x))\)-number.)