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 $ϕ$. We obtain two results bounding the exponent of $[G,ϕ]$. Denote by $G_{-ϕ}$ the set $\{[g,ϕ]\,\vert\, g\in G\}$ and by $G_ϕ$ the centralizer of $ϕ$, that is, the subgroup of fixed points of $ϕ$. 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 $x,y\in G_{-ϕ}$. Suppose that $G_ϕ$ is nilpotent of class $c$. Then the exponent of $[G,ϕ]$ is $(c,d,e)$-bounded.2. Assume that $G_ϕ$ has rank $r$ and $x^e=1$ for each $x\in G_{-ϕ}$. Then the exponent of $[G,ϕ]$ is $(e,r)$-bounded.