# Exponent of a finite group admitting a coprime automorphism

## TL;DR

This paper investigates bounds on the exponent of finite groups with coprime automorphisms, establishing conditions under which the group's exponent is bounded based on properties of centralizers and certain subsets.

## Contribution

It provides new bounds on the exponent of finite groups admitting coprime automorphisms, linking group structure and automorphism properties to exponent bounds.

## Key findings

- Exponent of G is bounded when elements are in invariant subgroups of bounded exponent.
- If G_phi is nilpotent and certain conditions hold, the exponent of [G,phi] is bounded.
- Results connect automorphism properties with group exponent bounds.

## Abstract

Let $G$ be a finite group admitting a coprime automorphism $\phi$ of order $n$. Denote by $G_{\phi}$ the centralizer of $\phi$ in $G$ and by $G_{-\phi}$ the set $\{ x^{-1}x^{\phi}; \ x\in G\}$. We prove the following results. 1. If every element from $G_{\phi}\cup G_{-\phi}$ is contained in a $\phi$-invariant subgroup of exponent dividing $e$, then the exponent of $G$ is $(e,n)$-bounded. 2. Suppose that $G_{\phi}$ is nilpotent of class $c$. If $x^{e}=1$ for each $x \in G_{-\phi}$ and any two elements of $G_{-\phi}$ are contained in a $\phi$-invariant soluble subgroup of derived length $d$, then the exponent of $[G,\phi]$ is bounded in terms of $c,d,e,n$.