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

Abstract

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

arXiv comment4 pages