Exponent of a finite group admitting a coprime automorphism

Abstract

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

arXiv comment11 pages