Coprime automorphisms of finite groups

Let $G$ be a finite group admitting a coprime automorphism $α$ of order $e$. Denote by $I_G(α)$ the set of commutators $g^{-1}g^α$, where $g\in G$, and by $[G,α]$ the subgroup generated by $I_G(α)$. We study the impact of $I_G(α)$ on the structure of $[G,α]$. Suppose that each subgroup generated by a subset of $I_G(α)$ can be generated by at most $r$ elements. We show that the rank of $[G,α]$ is $(e,r)$-bounded. Along the way, we establish several results of independent interest. In particular, we prove that if every element of $I_G(α)$ has odd order, then $[G,α]$ has odd order too. Further, if every pair of elements from $I_G(α)$ generates a soluble, or nilpotent, subgroup, then $[G,α]$ is soluble, or respectively nilpotent.

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