DOCSLIB.ORG

Automorphism Groups of Finite P-Groups: Structure and Applications

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Automorphism Groups of Finite P-Groups: Structure and Applications 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.
Load more

Recommended publications
  • Orbits of Automorphism Groups of Fields
    ORBITS OF AUTOMORPHISM GROUPS OF FIELDS KIRAN S. KEDLAYA AND BJORN POONEN Abstract. We address several specific aspects of the following general question: can a field K have so many automorphisms that the action of the automorphism group on the elements of K has relatively few orbits? We prove that any field which has only finitely many orbits under its automorphism group is finite. We extend the techniques of that proof to approach a broader conjecture, which asks whether the automorphism group of one field over a subfield can have only finitely many orbits on the complement of the subfield. Finally, we apply similar methods to analyze the field of Mal'cev-Neumann \generalized power series" over a base field; these form near-counterexamples to our conjecture when the base field has characteristic zero, but often fall surprisingly far short in positive characteristic. Can an infinite field K have so many automorphisms that the action of the automorphism group on the elements of K has only finitely many orbits? In Section 1, we prove that the answer is \no" (Theorem 1.1), even though the corresponding answer for division rings is \yes" (see Remark 1.2). Our proof constructs a \trace map" from the given field to a finite field, and exploits the peculiar combination of additive and multiplicative properties of this map. Section 2 attempts to prove a relative version of Theorem 1.1, by considering, for a non- trivial extension of fields k ⊂ K, the action of Aut(K=k) on K. In this situation each element of k forms an orbit, so we study only the orbits of Aut(K=k) on K − k.
    [Show full text]
  • The Pure Symmetric Automorphisms of a Free Group Form a Duality Group
    THE PURE SYMMETRIC AUTOMORPHISMS OF A FREE GROUP FORM A DUALITY GROUP NOEL BRADY, JON MCCAMMOND, JOHN MEIER, AND ANDY MILLER Abstract. The pure symmetric automorphism group of a finitely generated free group consists of those automorphisms which send each standard generator to a conjugate of itself. We prove that these groups are duality groups. 1. Introduction Let Fn be a finite rank free group with fixed free basis X = fx1; : : : ; xng. The symmetric automorphism group of Fn, hereafter denoted Σn, consists of those au- tomorphisms that send each xi 2 X to a conjugate of some xj 2 X. The pure symmetric automorphism group, denoted PΣn, is the index n! subgroup of Σn of symmetric automorphisms that send each xi 2 X to a conjugate of itself. The quotient of PΣn by the inner automorphisms of Fn will be denoted OPΣn. In this note we prove: Theorem 1.1. The group OPΣn is a duality group of dimension n − 2. Corollary 1.2. The group PΣn is a duality group of dimension n − 1, hence Σn is a virtual duality group of dimension n − 1. (In fact we establish slightly more: the dualizing module in both cases is -free.) Corollary 1.2 follows immediately from Theorem 1.1 since Fn is a 1-dimensional duality group, there is a short exact sequence 1 ! Fn ! PΣn ! OPΣn ! 1 and any duality-by-duality group is a duality group whose dimension is the sum of the dimensions of its constituents (see Theorem 9.10 in [2]). That the virtual cohomological dimension of Σn is n − 1 was previously estab- lished by Collins in [9].
    [Show full text]
  • Generalized Quaternions
    GENERALIZED QUATERNIONS KEITH CONRAD 1. introduction The quaternion group Q8 is one of the two non-abelian groups of size 8 (up to isomor- phism). The other one, D4, can be constructed as a semi-direct product: ∼ ∼ × ∼ D4 = Aff(Z=(4)) = Z=(4) o (Z=(4)) = Z=(4) o Z=(2); where the elements of Z=(2) act on Z=(4) as the identity and negation. While Q8 is not a semi-direct product, it can be constructed as the quotient group of a semi-direct product. We will see how this is done in Section2 and then jazz up the construction in Section3 to make an infinite family of similar groups with Q8 as the simplest member. In Section4 we will compare this family with the dihedral groups and see how it fits into a bigger picture. 2. The quaternion group from a semi-direct product The group Q8 is built out of its subgroups hii and hji with the overlapping condition i2 = j2 = −1 and the conjugacy relation jij−1 = −i = i−1. More generally, for odd a we have jaij−a = −i = i−1, while for even a we have jaij−a = i. We can combine these into the single formula a (2.1) jaij−a = i(−1) for all a 2 Z. These relations suggest the following way to construct the group Q8. Theorem 2.1. Let H = Z=(4) o Z=(4), where (a; b)(c; d) = (a + (−1)bc; b + d); ∼ The element (2; 2) in H has order 2, lies in the center, and H=h(2; 2)i = Q8.
    [Show full text]
  • On the Group-Theoretic Properties of the Automorphism Groups of Various Graphs
    ON THE GROUP-THEORETIC PROPERTIES OF THE AUTOMORPHISM GROUPS OF VARIOUS GRAPHS CHARLES HOMANS Abstract. In this paper we provide an introduction to the properties of one important connection between the theories of groups and graphs, that of the group formed by the automorphisms of a given graph. We provide examples of important results in graph theory that can be understood through group theory and vice versa, and conclude with a treatment of Frucht's theorem. Contents 1. Introduction 1 2. Fundamental Definitions, Concepts, and Theorems 2 3. Example 1: The Orbit-Stabilizer Theorem and its Application to Graph Automorphisms 4 4. Example 2: On the Automorphism Groups of the Platonic Solid Skeleton Graphs 4 5. Example 3: A Tight Bound on the Product of the Chromatic Number and Independence Number of Vertex-Transitive Graphs 6 6. Frucht's Theorem 7 7. Acknowledgements 9 8. References 9 1. Introduction Groups and graphs are two highly important kinds of structures studied in math- ematics. Interestingly, the theory of groups and the theory of graphs are deeply connected. In this paper, we examine one particular such connection: that which emerges from the observation that the automorphisms of any given graph form a group under composition. In section 2, we provide a framework for understanding the material discussed in the paper. In sections 3, 4, and 5, we demonstrate how important results in group theory illuminate some properties of automorphism groups, how the geo- metric properties of particular embeddings of graphs can be used to determine the structure of the automorphism groups of all embeddings of those graphs, and how the automorphism group can be used to determine fundamental truths about the structure of the graph.
    [Show full text]
  • The Centralizer of a Group Automorphism
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 54, 27-41 (1978) The Centralizer of a Group Automorphism CARLTON J. MAXSON AND KIRBY C. SWTH Texas A& M University, College Station, Texas 77843 Communicated by A. Fr6hlich Received May 31, 1971 Let G be a finite group. The structure of the near-ring C(A) of identity preserving functionsf : G + G, which commute with a given automorphism A of G, is investigated. The results are then applied to the case in which G is a finite vector space and A is an invertible linear transformation. Let A be a linear transformation on a finite-dimensional vector space V over a field F. The problem of determining the structure of the ring of linear trans- formations on V which commute with A has been studied extensively (e.g., [3, 5, 81). In this paper we consider a nonlinear analogue to this well-known problem which arises naturally in the study of automorphisms of a linear auto- maton [6]. Specifically let G be a finite group, written additively, and let A be an auto- morphism of G. If C(A) = {f: G + G j fA = Af and f (0) = 0 where 0 is the identity of G), then C(A) forms a near ring under the operations of pointwise addition and function composition. That is (C(A), +) is a group, (C(A), .) is a monoid, (f+-g)h=fh+gh for all f, g, hcC(A) and f.O=O.f=O, f~ C(A).
    [Show full text]
  • Homomorphisms and Isomorphisms
    Lecture 4.1: Homomorphisms and isomorphisms Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Modern Algebra M. Macauley (Clemson) Lecture 4.1: Homomorphisms and isomorphisms Math 4120, Modern Algebra 1 / 13 Motivation Throughout the course, we've said things like: \This group has the same structure as that group." \This group is isomorphic to that group." However, we've never really spelled out the details about what this means. We will study a special type of function between groups, called a homomorphism. An isomorphism is a special type of homomorphism. The Greek roots \homo" and \morph" together mean \same shape." There are two situations where homomorphisms arise: when one group is a subgroup of another; when one group is a quotient of another. The corresponding homomorphisms are called embeddings and quotient maps. Also in this chapter, we will completely classify all finite abelian groups, and get a taste of a few more advanced topics, such as the the four \isomorphism theorems," commutators subgroups, and automorphisms. M. Macauley (Clemson) Lecture 4.1: Homomorphisms and isomorphisms Math 4120, Modern Algebra 2 / 13 A motivating example Consider the statement: Z3 < D3. Here is a visual: 0 e 0 7! e f 1 7! r 2 2 1 2 7! r r2f rf r2 r The group D3 contains a size-3 cyclic subgroup hri, which is identical to Z3 in structure only. None of the elements of Z3 (namely 0, 1, 2) are actually in D3. When we say Z3 < D3, we really mean is that the structure of Z3 shows up in D3.
    [Show full text]
  • Automorphism Groups of Free Groups, Surface Groups and Free Abelian Groups
    Automorphism groups of free groups, surface groups and free abelian groups Martin R. Bridson and Karen Vogtmann The group of 2 × 2 matrices with integer entries and determinant ±1 can be identified either with the group of outer automorphisms of a rank two free group or with the group of isotopy classes of homeomorphisms of a 2-dimensional torus. Thus this group is the beginning of three natural sequences of groups, namely the general linear groups GL(n, Z), the groups Out(Fn) of outer automorphisms of free groups of rank n ≥ 2, and the map- ± ping class groups Mod (Sg) of orientable surfaces of genus g ≥ 1. Much of the work on mapping class groups and automorphisms of free groups is motivated by the idea that these sequences of groups are strongly analogous, and should have many properties in common. This program is occasionally derailed by uncooperative facts but has in general proved to be a success- ful strategy, leading to fundamental discoveries about the structure of these groups. In this article we will highlight a few of the most striking similar- ities and differences between these series of groups and present some open problems motivated by this philosophy. ± Similarities among the groups Out(Fn), GL(n, Z) and Mod (Sg) begin with the fact that these are the outer automorphism groups of the most prim- itive types of torsion-free discrete groups, namely free groups, free abelian groups and the fundamental groups of closed orientable surfaces π1Sg. In the ± case of Out(Fn) and GL(n, Z) this is obvious, in the case of Mod (Sg) it is a classical theorem of Nielsen.
    [Show full text]
  • Isomorphisms and Automorphisms
    Isomorphisms and Automorphisms Definition As usual an isomorphism is defined as a map between objects that preserves structure, for general designs this means: Suppose (X,A) and (Y,B) are two designs. They are said to be isomorphic if there exists a bijection α: X → Y such that if we apply α to the elements of any block of A we obtain a block of B and all blocks of B are obtained this way. Symbolically we can write this as: B = [ {α(x) | x ϵ A} : A ϵ A ]. The bijection α is called an isomorphism. Example We've seen two versions of a (7,4,2) symmetric design: k = 4: {3,5,6,7} {4,6,7,1} {5,7,1,2} {6,1,2,3} {7,2,3,4} {1,3,4,5} {2,4,5,6} Blocks: y = [1,2,3,4] {1,2} = [1,2,5,6] {1,3} = [1,3,6,7] {1,4} = [1,4,5,7] {2,3} = [2,3,5,7] {2,4} = [2,4,6,7] {3,4} = [3,4,5,6] α:= (2576) [1,2,3,4] → {1,5,3,4} [1,2,5,6] → {1,5,7,2} [1,3,6,7] → {1,3,2,6} [1,4,5,7] → {1,4,7,2} [2,3,5,7] → {5,3,7,6} [2,4,6,7] → {5,4,2,6} [3,4,5,6] → {3,4,7,2} Incidence Matrices Since the incidence matrix is equivalent to the design there should be a relationship between matrices of isomorphic designs.
    [Show full text]
  • Graph Automorphism Groups
    Graph Automorphism Groups Robert A. Beeler, Ph.D. East Tennessee State University February 23, 2018 Robert A. Beeler, Ph.D. (East Tennessee State University)Graph Automorphism Groups February 23, 2018 1 / 1 What is a graph? A graph G =(V , E) is a set of vertices, V , together with as set of edges, E. For our purposes, each edge will be an unordered pair of distinct vertices. a e b d c V (G)= {a, b, c, d, e} E(G)= {ab, ae, bc, be, cd, de} Robert A. Beeler, Ph.D. (East Tennessee State University)Graph Automorphism Groups February 23, 2018 2 / 1 Graph Automorphisms A graph automorphism is simply an isomorphism from a graph to itself. In other words, an automorphism on a graph G is a bijection φ : V (G) → V (G) such that uv ∈ E(G) if and only if φ(u)φ(v) ∈ E(G). Note that graph automorphisms preserve adjacency. In layman terms, a graph automorphism is a symmetry of the graph. Robert A. Beeler, Ph.D. (East Tennessee State University)Graph Automorphism Groups February 23, 2018 3 / 1 An Example Consider the following graph: a d b c Robert A. Beeler, Ph.D. (East Tennessee State University)Graph Automorphism Groups February 23, 2018 4 / 1 An Example (Part 2) One automorphism simply maps every vertex to itself. This is the identity automorphism. a a d b d b c 7→ c e =(a)(b)(c)(d) Robert A. Beeler, Ph.D. (East Tennessee State University)Graph Automorphism Groups February 23, 2018 5 / 1 An Example (Part 3) One automorphism switches vertices a and c.
    [Show full text]
  • Isomorphisms, Automorphisms, Homomorphisms, Kernels
    Isomorphisms, Automorphisms, Homomorphisms, Kernels September 10, 2009 Definition 1 Let G; G0 be groups and let f : G ! G0 be a mapping (of the underlying sets). We say that f is a (group) homomorphism if f(x · y) = f(x) · f(y) for all x; y 2 G. The composition f G−!G0−!h G" of two homomorphisms (as displayed) is again a homomorphism. For any two groups G, G0 by the trivial homomorphism from G to G0 we mean the mapping G ! G0 sending all elements to the identity element in G0. Recall: Definition 2 A group homomorphism f : G ! G0 is called a group isomorphism|or, for short, an isomorphism|if f is bijective. An isomorphism from a group G to itself is called an automorphism. Exercise 1 Let f : G ! G0 be a homomorphism of groups. The image under the homomorphism 0 f of any subgroup of G is a subgroup of G ; if H ⊂ G is generated by elements fx1; x2; : : : ; xng then its image f(H) is generated by the images ff(x1); f(x2); : : : ; f(xn)g; the image of any cyclic subgroup is cyclic; If f is an isomorphism from G to G0 and x 2 G then the order of x (which|by definition—is the order of the cyclic subgroup generated by x) is equal to the order of f(x). For G a group, let Aut(G) be the set of all automorphisms of G, with `composition of automorphisms" as its `composition law' (denoted by a center-dot (·). As noted in class, Proposition 1 If G is a group then Aut(G) is also a group.
    [Show full text]
  • Math 311: Complex Analysis — Automorphism Groups Lecture
    MATH 311: COMPLEX ANALYSIS | AUTOMORPHISM GROUPS LECTURE 1. Introduction Rather than study individual examples of conformal mappings one at a time, we now want to study families of conformal mappings. Ensembles of conformal mappings naturally carry group structures. 2. Automorphisms of the Plane The automorphism group of the complex plane is Aut(C) = fanalytic bijections f : C −! Cg: Any automorphism of the plane must be conformal, for if f 0(z) = 0 for some z then f takes the value f(z) with multiplicity n > 1, and so by the Local Mapping Theorem it is n-to-1 near z, impossible since f is an automorphism. By a problem on the midterm, we know the form of such automorphisms: they are f(z) = az + b; a; b 2 C; a 6= 0: This description of such functions one at a time loses track of the group structure. If f(z) = az + b and g(z) = a0z + b0 then (f ◦ g)(z) = aa0z + (ab0 + b); f −1(z) = a−1z − a−1b: But these formulas are not very illuminating. For a better picture of the automor- phism group, represent each automorphism by a 2-by-2 complex matrix, a b (1) f(z) = ax + b ! : 0 1 Then the matrix calculations a b a0 b0 aa0 ab0 + b = ; 0 1 0 1 0 1 −1 a b a−1 −a−1b = 0 1 0 1 naturally encode the formulas for composing and inverting automorphisms of the plane. With this in mind, define the parabolic group of 2-by-2 complex matrices, a b P = : a; b 2 ; a 6= 0 : 0 1 C Then the correspondence (1) is a natural group isomorphism, Aut(C) =∼ P: 1 2 MATH 311: COMPLEX ANALYSIS | AUTOMORPHISM GROUPS LECTURE Two subgroups of the parabolic subgroup are its Levi component a 0 M = : a 2 ; a 6= 0 ; 0 1 C describing the dilations f(z) = ax, and its unipotent radical 1 b N = : b 2 ; 0 1 C describing the translations f(z) = z + b.
    [Show full text]
  • Automorphisms of Models of Set Theory and NFU
    Automorphisms of Models of Set Theory and NFU Zach McKenzie I A Brief Introductuion to NFU I The connection between models of ZFC that admit automorphism and NFU I Some models that admit automorphism I The connection between models of ZFC that admit automorphism and NFU I Some models that admit automorphism I A Brief Introductuion to NFU I Some models that admit automorphism I A Brief Introductuion to NFU I The connection between models of ZFC that admit automorphism and NFU I A Brief Introductuion to NFU I The connection between models of ZFC that admit automorphism and NFU I Some models that admit automorphism Definition We say that an L-formula φ is stratified if and only if there is a function σ from the variables appearing φ to the natural numbers such that (i) if x 2 y is a subformula of φ then σ(y) = σ(x) + 1, (ii) if x = y is a subformula of φ then σ(x) = σ(y). I The class of stratified formulae is exactly the class of well-formed formulae of Russell's Theory of Types with the type indices removed. Stratified Formulae I Let L = hS; 2i where S is a unary predicate and 2 a binary predicate. I The class of stratified formulae is exactly the class of well-formed formulae of Russell's Theory of Types with the type indices removed. Stratified Formulae I Let L = hS; 2i where S is a unary predicate and 2 a binary predicate. Definition We say that an L-formula φ is stratified if and only if there is a function σ from the variables appearing φ to the natural numbers such that (i) if x 2 y is a subformula of φ then σ(y) = σ(x) + 1, (ii) if x = y is a subformula of φ then σ(x) = σ(y).
    [Show full text]