A characteristic subgroup of \({\mathcal N}\)-stable groups
The aim of this paper is to obtain factorization theorems for certain finite groups like the well-known Thompson-Glaubermann results. To describe one of them, let \(\tilde {\mathcal N}=\{G|\) \(G=F^*(G)\}\) denote the class of quasinilpotent groups as introduced by Bender. By a result of \textit{D. Blessenohl} and \textit{H. Laue} [J. Algebra 56, 516-532 (1979; Zbl 0416.20015)], every finite group G possesses a unique conjugacy class of \(\tilde {\mathcal N}\)-injectors; these are just the \(\tilde {\mathcal N}\)-maximal subgroups of G containing \(F^*(G)\). Then, if G is an \({\mathcal N}\)-stable group (i.e. \(B\leq Fit(N_ G(A) mod C_ G(A))\) for all nilpotent subgroups A of G and B of \(N_ G(A)\) with \([A,B,B]=1)\) such that \(1\neq Fit(G)\) is not a 2-group and if K is an \(\tilde {\mathcal N}\)-injector of G, \(O_{2'}(Z(J(K))\) is a non-identity normal subgroup of G. (Here J(K) denotes the Thompson-subgroup of K, defined as the group generated by all abelian subgroups of K of maximal order.) It follows that \[ G=N_ G(J(K))\cdot C_ G(O_{2'}(Z(J(K)))=N_ G(J(K))\cdot C_ G(O_{2'}(Z(K)). \]
conjugacy class of \(\tilde {\mathcal N}\)-injectors
\(\tilde {\mathcal N}\)-maximal subgroups
scientific article; zbMATH DE number 3971035