Coprime automorphisms of finite groups

TL;DR

This paper investigates how coprime automorphisms influence the structure of finite groups, establishing bounds on the rank of certain subgroups and deriving properties like solubility and nilpotency under specific conditions.

Contribution

It introduces bounds on the rank of the subgroup generated by commutators related to coprime automorphisms and proves new results on the structure and properties of these subgroups.

Findings

The rank of [G,α] is bounded by (e, r).

If all elements of I_G(α) have odd order, then [G,α] has odd order.

Pairs of elements generating soluble or nilpotent subgroups imply [G,α] is soluble or nilpotent.

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.