Intense automorphisms of finite groups

TL;DR

This paper studies intense automorphisms of finite groups, especially finite p-groups, classifying those with non-p-group automorphism groups and revealing their structure related to nilpotency class.

Contribution

It classifies finite p-groups with non-p-group intense automorphism groups and links their structure to nilpotency class and pro-p groups.

Findings

Automorphism group structure depends on nilpotency class.

Finite p-groups with non-p-group intense automorphisms are classified.

Structure related to infinite pro-p groups.

Abstract

Let $G$ be a group. An automorphism of $G$ is called intense if it sends each subgroup of $G$ to a conjugate; the collection of such automorphisms is denoted by $\mathrm{Int}(G)$. In the special case in which $p$ is a prime number and $G$ is a finite $p$-group, one can show that $\mathrm{Int}(G)$ is the semidirect product of a normal $p$-Sylow and a cyclic subgroup of order dividing $p-1$. In this thesis we classify the finite $p$-groups whose groups of intense automorphisms are not themselves $p$-groups. It emerges from our investigation that the structure of such groups is almost completely determined by their nilpotency class: for $p>3$, they share a quotient, growing with their class, with a uniquely determined infinite $2$-generated pro-$p$ group.