Noncoprime action of a cyclic group

Abstract

Let A be a finite nilpotent group acting fixed point freely on the finite (solvable) group G by automorphisms. It is conjectured that the nilpotent length of G is bounded above by l(A), the number of primes dividing the order of A counted with multiplicities. In the present paper we consider the case A is cyclic and obtain that the nilpotent length of G is at most 2l(A) if vertical bar G vertical bar is odd. More generally we prove that the nilpotent length of G is at most 2l(A) + c(G; A) when G is of odd order and A normalizes a Sylow system of G where c(G; A) denotes the number of trivial A-modules appearing in an A-composition series of G. (c) 2023 Elsevier Inc. All rights reserved.