Finite groups with a large automorphism orbit

Abstract

We study the nonabelian composition factors of a finite group $G$ assumed to admit an $\mathrm{Aut}(G)$-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 $\mathrm{Aut}(G)$-orbit of length at least $c(S)|G|$.

35 pages