Finite Groups with 6 or 7 Automorphism Orbits

TL;DR

This paper classifies finite nonsolvable groups with up to 6 automorphism orbits, proves infinitely many have exactly 7, and generalizes finiteness results for groups with bounded automorphism orbits.

Contribution

It provides a classification of finite nonsolvable groups with at most 6 automorphism orbits and extends finiteness results to groups with bounded automorphism orbits.

Findings

Finite nonsolvable groups with ≤6 automorphism orbits are classified.

Infinitely many finite nonsolvable groups have exactly 7 automorphism orbits.

Finitely many groups without nontrivial abelian normal subgroups have bounded automorphism orbits.

Abstract

Let $G$ be a group. The orbits of the natural action of $\mbox{Aut}(G)$ on $G$ are called "automorphism orbits" of $G$, and the number of automorphism orbits of $G$ is denoted by $\omega(G)$. In this paper the finite nonsolvable groups $G$ with $\omega(G) \leq 6$ are classified - this solves a problem posed by Markus Stroppel - and it is proved that there are infinitely many finite nonsolvable groups $G$ with $\omega(G)=7$. Moreover it is proved that for a given number $n$ there are only finitely many finite groups $G$ without nontrivial abelian normal subgroups and such that $\omega(G) \leq n$, generalizing a result of Kohl.