Reduction theorems for characteristic functors on finite \(p\)-groups and applications to \(p\)-nilpotence criteria. (Q1001970)

The main result of the paper, Theorem 4.1, establishes various implications between six properties of characteristic \(p\)-functors. A factorization theorem for a 2-constrained, \(S_4\)-free finite group with a Sylow 2-subgroup \(S\) of nilpotency class at most two is given in Theorem 5.1: \(G=N_G(W(S))O_{2'}(G)\), where \(W(S)\) is a characteristic 2-functor introduced by \textit{G. Glauberman} [Factorizations in local subgroups of finite groups. Conference Board of the Mathematical Sciences. Regional Conference Series in Mathematics 33. Providence: AMS (1977; Zbl 0489.20012)].