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.
Categories: math.GR