Length-type parameters of finite groups with almost unipotent automorphisms

Abstract

Let α be an automorphism of a finite group G. For a positive integer n, let E_G,n(α) be the subgroup generated by all commutators [...[[x,α],α],…,α] in the semidirect product G 〈α〉 over x ∈ G, where α is repeated n times. By Baer’s theorem, if E_G,n(α)=1, then the commutator subgroup [G,α] is nilpotent. We generalize this theorem in terms of certain length parameters of E_G,n(α). For soluble G we prove that if, for some n, the Fitting height of E_G,n(α) is equal to k, then the Fitting height of [G,α] is at most k + 1. For nonsoluble G the results are in terms of the nonsoluble length and generalized Fitting height. The generalized Fitting height h*(H) of a finite group H is the least number h such that F_h* (H) = H, where F₀* (H) = 1, and F_i+1* (H) is the inverse image of the generalized Fitting subgroup F*(H/F_i*(H)). Let m be the number of prime factors of the order |α| counting multiplicities. It is proved that if, for some n, the generalized Fitting height E_G,n(α) of is equal to k, then the generalized Fitting height of [G,α] is bounded in terms of k and m. The nonsoluble length λ(H) of a finite group H is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if λE_G,n(α)= k, then the nonsoluble length of [G,α] is bounded in terms of k and m. We also state conjectures of stronger results independent of m and show that these conjectures reduce to a certain question about automorphisms of direct products of finite simple groups.