Automorphism Groups of Finite p-Groups: Structure and Applications

Details Automorphism Groups of Finite p-Groups: Structure and Applications Automorphism Groups of Finite p-Groups: Structure and Applications

2007-11-18

arxiv.org/abs/0711.2816v1

Mathematics

Group Theory

107 pages, Ph.D. thesis

Scientific paper

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.

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