Exponent of a finite group admitting a coprime automorphism

Abstract

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