Finite groups with a large automorphism orbit

TL;DR

This paper investigates finite groups with large automorphism orbits, showing bounds on their nonabelian composition factors and conditions for solvability, while also constructing groups with specific simple composition factors and large automorphism orbits.

Contribution

It establishes bounds on nonabelian composition factors based on automorphism orbit size and characterizes solvability for large orbit sizes, also constructing groups with prescribed simple factors.

Findings

Nonabelian composition factors are bounded in size by the automorphism orbit proportion.

Groups with orbit size > 18/19 of the group are solvable.

Existence of groups with specific simple factors and arbitrarily large automorphism orbits.

Abstract

We study the nonabelian composition factors of a finite group $G$ assumed to admit an $\operatorname{Aut}(G)$-orbit of length at least $\rho|G|$, for a given $\rho\in\left(0,1\right]$. Our main results are the following: The orders of the nonabelian composition factors of $G$ are then bounded in terms of $\rho$, and if $\rho>\frac{18}{19}$, then $G$ is solvable. On the other hand, for each nonabelian finite simple group $S$, there is a constant $c(S)\in\left(0,1\right]$ such that $S$ occurs with arbitrarily large multiplicity as a composition factor in some finite group $G$ having an $\operatorname{Aut}(G)$-orbit of length at least $c(S)|G|$.