Classification of finite group automorphisms with a large cycle

Abstract

Let $ψ$ be a permutation of a finite set $X$. We define $λ(ψ)$ to be the largest fraction of elements of $X$ lying on a single cycle of $ψ$. For a finite group $G$, we define $λ(G)$ to be the maximum among the values $λ(α)$, where $α$ runs through the automorphisms of $G$. In this paper, we develop tools to deal with questions related to $λ$-values of finite groups and of their automorphisms. As a consequence, we will be able to give a classification, up to a natural notion of isomorphism, of those pairs $(G,α)$ where $G$ is a finite group, $α$ is an automorphism of $G$ and $λ(α)\geq\frac{1}{2}$.