Coprime automorphisms of finite groups

Abstract

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

Final version to appear in Transactions of the American Mathematical Society