# Exponent of a finite group of odd order with an involutory automorphism

## TL;DR

This paper establishes bounds on the exponent of the subgroup generated by elements of the form [g,φ] in a finite odd-order group with an involutory automorphism, linking these bounds to properties of the fixed point subgroup.

## Contribution

It provides new bounds on the exponent of [G,φ] based on the structure and properties of the fixed point subgroup G_φ and elements in G_{-φ}, extending understanding of automorphism actions.

## Key findings

- Bound on exponent when G_φ is nilpotent of class c
- Bound on exponent when G_φ has rank r
- Results depend on properties of elements in G_{-φ}

## Abstract

Let $G$ be a finite group of odd order admitting an involutory automorphism $\phi$. We obtain two results bounding the exponent of $[G,\phi]$. Denote by $G_{-\phi}$ the set $\{[g,\phi]\,\vert\, g\in G\}$ and by $G_{\phi}$ the centralizer of $\phi$, that is, the subgroup of fixed points of $\phi$. The obtained results are as follows.1. Assume that the subgroup $\langle x,y\rangle$ has derived length at most $d$ and $x^e=1$ for every $x,y\in G_{-\phi}$. Suppose that $G_\phi$ is nilpotent of class $c$. Then the exponent of $[G,\phi]$ is $(c,d,e)$-bounded.2. Assume that $G_\phi$ has rank $r$ and $x^e=1$ for each $x\in G_{-\phi}$. Then the exponent of $[G,\phi]$ is $(e,r)$-bounded.