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