\({\mathcal F}\)-stability of finite groups (Q801153)

Given a saturated formation \({\mathcal F}\) of finite solvable groups, the author calls a finite group G \({\mathcal F}\)-stable if whenever H is an \({\mathcal F}\)-subgroup of G and \(x\in N_ G(H)\) such that \([H,x,x]=1\), then \(xC_ G(H)\) is contained in the group generated by all subnormal \({\mathcal F}\)-subgroups of \(N_ G(H)/C_ G(H)\). It is proved that the following generalization of a theorem due to Glauberman is valid: If \({\mathcal F}\) has odd characteristic then every section of G is \({\mathcal F}\)-stable if and only if no section of G is isomorphic to a special affine group SA(2,p), \(p\in char {\mathcal F}\). Consequently, for saturated formations of odd characteristic, \({\mathcal F}\)-stability is nothing else than p-stability for all \(p\in char {\mathcal F}\).