Automorphism Groups of Finite p-Groups: Structure and Applications

TL;DR

This thesis investigates the structure of automorphism groups of finite p-groups, proving that almost all such groups have automorphism groups that are p-groups, and explores applications including Markov chain convergence.

Contribution

It provides a complete proof of a theorem about automorphism groups of finite p-groups, offers a comprehensive survey of existing results, and connects automorphisms to Markov chain analysis.

Findings

Almost all finite p-groups have automorphism groups that are p-groups.

The thesis includes a detailed proof and exposition of the main theorem.

A novel connection between automorphisms of p-groups and Markov chain convergence rates.

Abstract

This thesis has three goals related to the automorphism groups of finite $p$-groups. The primary goal is to provide a complete proof of a theorem showing that, in some asymptotic sense, the automorphism group of almost every finite $p$-group is itself a $p$-group. We originally proved this theorem in a paper with Martin; the presentation of the proof here contains omitted proof details and revised exposition. We also give a survey of the extant results on automorphism groups of finite $p$-groups, focusing on the order of the automorphism groups and on known examples. Finally, we explore a connection between automorphisms of finite $p$-groups and Markov chains. Specifically, we define a family of Markov chains on an elementary abelian $p$-group and bound the convergence rate of some of those chains.