Noncoprime action of a cyclic group

TL;DR

This paper investigates the nilpotent length of a finite solvable group acted upon by a cyclic group, establishing bounds related to the prime factorization of the acting group's order and the structure of the group.

Contribution

It proves new bounds on the nilpotent length of groups under cyclic automorphism actions, extending previous conjectures to specific cases with odd order groups.

Findings

Nilpotent length is at most 2 times the number of primes dividing the cyclic group order for odd order groups.

Introduces a bound involving the number of trivial modules in an A-composition series.

Establishes results when the group normalizes a Sylow system, linking structure to automorphism actions.

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 $\ell(A)$, 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 $2\ell(A)$ if $|G|$ is odd. More generally we prove that the nilpotent length of $G$ is at most $2\ell(A)+ \mathbf{c}(G;A)$ when $G$ is of odd order and $A$ normalizes a Sylow system of $G$ where $\mathbf{c}(G;A)$ denotes the number of trivial $A$-modules appearing in an $A$-composition series of $G$.