On the influence of fixed point free nilpotent automorphism groups

Abstract

A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F with a nontrivial complement H such that \( [F,h]=F\) for all nonidentity elements \(h\in H\). Let FH be a Frobenius-like group with complement H of prime order such that \(C_F(H)\) is of prime order. Suppose that FH acts on a finite group G by automorphisms where \( (|G|,|H|)=1\) in such a way that \(C_G(F)=1.\) In the present paper we prove that the Fitting series of \(C_G(H)\) coincides with the intersections of \( C_G(H)\) with the Fitting series of G, and the nilpotent length of G exceeds the nilpotent length of \(C_G(H)\) by at most one. As a corollary, we also prove that for any set of primes \(\pi \), the upper \(\pi \)-series of \( C_G(H)\) coincides with the intersections of \(C_G(H)\) with the upper \(\pi \) -series of G, and the \(\pi \)- length of G exceeds the \(\pi \)-length of \( C_G(H)\) by at most one.