On finite groups with few automorphism orbits

TL;DR

This paper classifies finite nonsolvable groups with a small number of automorphism orbits, showing they are isomorphic to specific well-known groups when the orbit count is up to 6.

Contribution

It provides a complete classification of nonsolvable finite groups with at most 6 automorphism orbits, identifying specific isomorphism types and structural conditions.

Findings

Groups with ≤5 automorphism orbits are isomorphic to A_5, A_6, PSL(2,7), or PSL(2,8)

For exactly 6 orbits, groups are either PSL(3,4) or have a characteristic elementary abelian 2-subgroup with a quotient isomorphic to A_5

The results narrow down the structure of groups with few automorphism orbits

Abstract

Denote by $\omega(G)$ the number of orbits of the action of $Aut(G)$ on the finite group $G$. We prove that if $G$ is a finite nonsolvable group in which $\omega(G) \leqslant 5$, then $G$ is isomorphic to one of the groups $A_5,A_6,PSL(2,7)$ or $PSL(2,8)$. We also consider the case when $\omega(G) = 6$ and show that if $G$ is a nonsolvable finite group with $\omega(G) = 6$, then either $G \simeq PSL(3,4)$ or there exists a characteristic elementary abelian $2$-subgroup $N$ of $G$ such that $G/N \simeq A_5$.