Automorphism Groups of Finite p-Groups: Structure and Applications A Dissertation Submitted to the Department of Mathematics at Stanford University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Geir T. Helleloid Department of Mathematics The University of Texas at Austin 1 University Station C1200 Austin, TX 78712 Current Work Email Address: [email protected] Permanent Email Address: [email protected] arXiv:0711.2816v1 [math.GR] 18 Nov 2007 August 2007 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. Acknowledgments First, I would like to thank my advisor, Persi Diaconis. Over the past five years, he has shown me the beautiful connections between random walks, combinatorics, finite group theory, and plenty of other mathematics. His continued encouragement and advice made the completion of this thesis possible. Thanks also go to Dan Bump, Nat Thiem, and Ravi Vakil for being on my defense committee and helping me finish my last hurdle as a Ph.D. student. My mentor Joe Gallian has influenced my life in more ways than I ever could have imagined when I first went to his REU in the summer of 2001. Spending one summer in Duluth as a student, two summers as a research advisor, and four summers as a research visitor has done more for my development as a research mathematician, teacher, and mentor than anything else in my life. At the same time, I first met most of my best friends at Duluth, and I will always thank Joe for bringing Phil Matchett Wood, Melanie Wood, Dan Isaksen, David Arthur, Stephen Hartke, and so many others into my life. I would also like to thank my friends at Stanford for their steadfast friendship over the years, particularly Leo Rosales, Dan Ramras, and Dana Paquin. I’ve saved the best for last. My parents and my girlfriend Jenny are the three most important people in my life, and I cannot thank them enough for their unconditional love and support. Jenny was my cheerleader in the last stressful year of my graduate studies, and my parents have been my cheerleaders for the last 26 years. Thank you. 2 Contents 1 Introduction 7 2 The Automorphism Group is Almost Always a p-Group 11 2.1 ExamplesandComputationalData . ... 11 2.2 TheMainTheorem................................ 13 2.3 AnOutlineoftheProofoftheMainTheorem . .... 14 2.4 Concluding Remarks on the Main Theorem . .... 17 3 The Lower p-Series and the Enumeration of p-Groups 21 3.1 The Lower p-Series ................................ 21 3.2 The Lower p-SeriesandAutomorphisms . 23 3.3 EnumeratingGroupsinaVariety . ... 24 4 The Lower p-Series of a Free Group 29 4.1 TheFreeLieAlgebra............................... 29 4.2 The Free Lie Algebra and Fn/Fn + 1 ....................... 30 4.3 The Expansion of Subgroups of F/Fn + 1 .................... 36 5 Counting Normal Subgroups of Finite p-Groups 41 5.1 The Number of Normal Subgroups of a Finite p-Group ............ 41 5.2 AProofofTheorem2.4 ............................. 44 6 Counting Submodules 47 6.1 The Number of Submodules of a Module . .. 47 6.2 AProofofTheorem2.6 ............................. 54 7 A Survey on the Automorphism Groups of Finite p-Groups 57 7.1 The Automorphisms of Familiar p-Groups ................... 57 7.1.1 The Extraspecial p-Groups........................ 57 7.1.2 Maximal Unipotent Subgroups of a Chevalley Group . ..... 60 7.1.3 Sylow p-SubgroupsoftheSymmetricGroup . 63 7.1.4 p-GroupsofMaximalClass . 63 7.1.5 Stem Covers of an Elementary Abelian p-Group ............ 64 7.2 QuotientsofAutomorphismGroups. .... 64 3 7.2.1 The Quotient Aut(G)/Autc(G) ..................... 65 7.2.2 The Quotient Aut(G)/Autf (G) ..................... 66 7.3 OrdersofAutomorphismGroups . .. 66 7.3.1 NilpotentAutomorphismGroups . 67 7.3.2 WreathProducts ............................. 67 7.3.3 The Automorphism Group of an Abelian p-Group........... 68 7.3.4 Other p-Groups Whose Automorphism Groups are p-Groups . 68 8 An Application of Automorphisms of p-Groups 71 8.1 ATwistedMarkovChain............................. 71 8.1.1 An Upper Bound on the Convergence Rate . 73 8.1.2 ALowerBoundontheConvergenceRate . 76 A Numerical Estimates for Theorem 2.1 79 B Numerical Estimates for Theorem 8.1 89 4 Notation and Terminology Let G be a group. If x, y G, then [x, y]= x−1y−1xy. • ∈ G′ = [G,G]= [x, y] : x, y G • h ∈ i Z(G) = the center of G • Φ(G) = the Frattini subgroup of G • Inn(G) = the group of inner automorphisms of G • Out(G) = Aut(G)/Inn(G) = the group of outer automorphisms of G • C = the cyclic group on n elements • n d(G) = the minimum cardinality of a generating set of G. If G is a free group, then • d(G) is called the rank of G. If G is a finite elementary abelian p-group, then d(G) is the dimension of G as an Fp-vector space and thus is called the dimension of G. GL(d, F ) = the general linear group of dimension d over the finite field F • q q n = the Gaussian (or q-binomial) coefficent. This equals the number of k-dimensional • k q subspaces of an F -vector space of dimension n. q (q) = the Galois number. This equals the total number of subspaces of an F -vector • Gn q space of dimension n. 5 6 Chapter 1 Introduction For any fixed prime p, a non-trivial group G is a p-group if the order of every element of G is a power of p. When G is finite, this is equivalent to saying that the order of G is a power of p. The study of p-groups (and particularly finite p-groups) is an important subfield of group theory. One motivation for studying finite p-groups is Sylow’s Theorem 1, which states that if G is a finite group, p divides the order of G, and pn is the largest power of p dividing the order of G, then G has at least one subgroup of order pn. Such subgroups are called the Sylow p-subgroups of G. The fact that G has at least one Sylow p-subgroup for each prime p dividing the order of G suggests that in some heuristic sense, finite p-groups are the “building blocks” of finite groups, and that to understand finite groups, we must first understand finite p-groups. This thesis studies the automorphism groups of finite p-groups, but this introductory chapter begins with a description of two other aspects of finite p-group theory, both to give a sense for what p-group theorists study, and because they are relevant to the main question addressed in this thesis. It turns out that understanding finite p-groups (whatever that means) is quite hard. Mann [68] has a wonderful survey of research and open questions in p-group theory. Much of the research relies on a basic fact about finite p-groups: they are nilpotent groups. A group G is nilpotent if the series of subgroups H0 = G, H1 = [G,G], H2 = [H1,G], . , eventually reaches the trivial subgroup. Here, [Hi,G] denotes the subgroup of G generated by all commutators consisting of an element in Hi and an element in G. If m is the smallest positive integer such that Hm is trivial, then we say that G is nilpotent of class m, or just of class m. When G is a finite p-group and the order of G is pn, the class of G is at least 1 and at most n 1. Finite p-groups of class 1 are the abelian p-groups, and those of class n 1 − − are said to be of maximal class. One triumph in finite p-group theory over the past 30 years has been the positive res- olution of the five coclass conjectures via the joint efforts of several researchers. While we have no hope of a complete classification of finite p-groups up to isomorphism (see Leedham- Green and McKay [59, Preface]), the coclass conjectures do offer a lot of information about finite p-groups. Mann [68, Section 3] has a short discussion of the subject, and the book by Leedham-Green and McKay [59] is devoted to a proof of the conjectures and related research. We will state only one of the conjectures here. The coclass of a finite p-group of 7 order pn and class m is defined to be n m. The coclass Conjecture A states that for some function f(p,r), every finite p-group of− coclass r has a normal subgroup K of class at most 2 and index at most f(p,r). If p = 2, one can require K to be abelian. Another aspect of p-group theory is the enumeration of finite p-groups by their order and related questions, as described in Mann [68, Section 1]. Let g(k) equal the number of groups of order at most k, let gnil(k) equal the number of nilpotent groups of order at most k, let gp(k) equal the number of p-groups of order at most k, and let gp,2(k) equal the number of p-groups of order at most k and class 2.