Classifying finite groups G with three Aut(G)-orbits

TL;DR

This paper classifies all finite groups where the automorphism group action results in exactly three orbits, providing a complete list including several infinite families and recent classifications of specific non-abelian p-groups.

Contribution

It offers a comprehensive, irredundant classification of finite groups with exactly three automorphism orbits, including new classifications of certain non-abelian p-groups.

Findings

Seven infinite families identified, including abelian and non-nilpotent groups.

Complete classification of non-abelian 2-groups with three automorphism orbits.

Recent independent classifications of non-abelian p-groups with p odd included.

Abstract

We give a complete and irredundant list of the finite groups $G$ for which Aut$(G)$, acting naturally on $G$, has precisely $3$ orbits. There are 7 infinite families: one abelian, one non-nilpotent, three families of non-abelian $2$-groups and two families of non-abelian $p$-groups with $p$ odd. The non-abelian $2$-group examples were first classified by Bors and Glasby in 2020 and non-abelian $p$-group examples with $p$ odd were classified independently by Li and Zhu, and by the author, in March 2024.