2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/142392This 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.107 pages, Ph.D. thesisGroup TheoryProbability20E36text