Finite groups with only small automorphism orbits

TL;DR

This paper classifies finite groups with automorphism orbits of bounded size, showing specific small groups have orbits of size at most 3, infinitely many have orbits up to size 8, and larger groups are constrained by bounds related to their generators and solvability.

Contribution

The paper provides a complete classification for groups with orbits of size at most 3, constructs examples with orbits up to size 8, and establishes bounds on group order based on orbit size and number of generators.

Findings

Groups with orbit size ≤ 3 are explicitly classified.

Infinitely many groups have orbit size exactly 8.

Groups with orbit size ≤ 23 are solvable.

Abstract

We study finite groups $G$ such that the maximum length of an orbit of the natural action of the automorphism group $\operatorname{Aut}(G)$ on $G$ is bounded from above by a constant. Our main results are the following: Firstly, a finite group $G$ only admits $\operatorname{Aut}(G)$-orbits of length at most $3$ if and only if $G$ is cyclic of one of the orders $1$, $2$, $3$, $4$ or $6$, or $G$ is the Klein four group or the symmetric group of degree $3$. Secondly, there are infinitely many finite ($2$-)groups $G$ such that the maximum length of an $\operatorname{Aut}(G)$-orbit on $G$ is $8$. Thirdly, the order of a $d$-generated finite group $G$ such that $G$ only admits $\operatorname{Aut}(G)$-orbits of length at most $c$ is explicitly bounded from above in terms of $c$ and $d$. Fourthly, a finite group $G$ such that all $\operatorname{Aut}(G)$-orbits on $G$ are of length at most $23$ is solvable.