DOCSLIB.ORG

Group Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Group Theory Group Theory J.S. Milne S3 Â1 2 3à r D 2 3 1 Â1 2 3à f D 1 3 2 Version 4.00 June 23, 2021 The first version of these notes was written for a first-year graduate algebra course.Asin most such courses, the notes concentrated on abstract groups and, in particular, on finite groups. However, it is not as abstract groups that most mathematicians encounter groups, but rather as algebraic groups, topological groups, or Lie groups, and it is not just the groups themselves that are of interest, but also their linear representations. It is my intention (one day) to expand the notes to take account of this, and to produce a volume that, while still modest in size (c200 pages), will provide a more comprehensive introduction to group theory for beginning graduate students in mathematics, physics, and related fields. BibTeX information @misc{milneGT, author={Milne, James S.}, title={Group Theory (v4.00)}, year={2021}, note={Available at www.jmilne.org/math/}, pages={139} } Please send comments and corrections to me at jmilne at umich dot edu. v2.01 (August 21, 1996). First version on the web; 57 pages. v2.11 (August 29, 2003). Revised and expanded; numbering; unchanged; 85 pages. v3.00 (September 1, 2007). Revised and expanded; 121 pages. v3.16 (July 16, 2020). Revised and expanded; 137 pages. v4.00 (June 23, 2021). Made document (including source code) available under Creative Commons licence. The multiplication table of S3 on the front page was produced by Group Explorer. Version 4.0 is published under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International licence (CC BY-NC-SA 4.0). Licence information: https://creativecommons.org/licenses/by-nc-sa/4.0 Copyright ©1996–2021 J.S. Milne. Contents Contents3 1 Basic Definitions and Results7 Definitions and examples.............................. 7 Multiplication tables ................................ 11 Subgroups...................................... 12 Groups of small order................................ 15 Homomorphisms .................................. 16 Cosets........................................ 17 Normal subgroups.................................. 18 Kernels and quotients................................ 20 Theorems concerning homomorphisms....................... 21 Direct products ................................... 23 Commutative groups ................................ 25 The order of ab ................................... 28 Exercises ...................................... 29 2 Free Groups and Presentations; Coxeter Groups 31 Free monoids .................................... 31 Free groups ..................................... 32 Generators and relations .............................. 35 Finitely presented groups.............................. 37 Coxeter groups ................................... 38 Exercises ...................................... 41 3 Automorphisms and Extensions 43 Automorphisms of groups ............................. 43 Characteristic subgroups .............................. 45 Semidirect products................................. 46 Extensions of groups ................................ 50 The Holder¨ program. ................................ 52 Exercises ...................................... 54 4 Groups Acting on Sets 57 Definition and examples .............................. 57 Permutation groups................................. 64 The Todd-Coxeter algorithm............................. 70 Primitive actions. .................................. 71 Exercises ...................................... 73 5 The Sylow Theorems; Applications 75 3 The Sylow theorems ................................ 75 Alternative approach to the Sylow theorems.................... 79 Examples ...................................... 80 Exercises ...................................... 83 6 Subnormal Series; Solvable and Nilpotent Groups 85 Subnormal Series................................... 85 Solvable groups................................... 88 Nilpotent groups .................................. 91 Groups with operators................................ 94 Krull-Schmidt theorem............................... 96 Exercises ...................................... 97 7 Representations of Finite Groups 99 Matrix representations ............................... 99 Roots of 1 in fields .................................100 Linear representations................................100 Maschke’s theorem.................................101 The group algebra; semisimplicity .........................103 Semisimple modules ................................104 Simple F -algebras and their modules........................106 Semisimple F -algebras and their modules.....................110 The representations of G ..............................112 The characters of G .................................113 The character table of a group ...........................116 Examples ......................................116 Exercises ......................................116 A Additional Exercises 119 B Solutions to the Exercises 123 C Two-Hour Examination 131 Bibliography 133 Index 135 4 NOTATION. We use the standard (Bourbaki) notation: N 0;1;2;::: ; Z is the ring of integers; Q is the D f g field of rational numbers; R is the field of real numbers; C is the field of complex numbers; Fq is a finite field with q elements, where q is a power of a prime number. In particular, Fp Z=pZ for p a prime number. D For integers m and n, m n means that m divides n, i.e., n mZ. Throughout the notes, j 2 p is a prime number, i.e., p 2;3;5;7;11;:::;1000000007;:::. D Given an equivalence relation, Œ denotes the equivalence class containing . The empty set is denoted by . The cardinality of a set S is denoted by S (so S is the number of ; j j j j elements in S when S is finite). Let I and A be sets; a family of elements of A indexed by 1 I , denoted .ai /i I , is a function i ai I A. 2 7! W ! Rings are required to have an identity element 1, and homomorphisms of rings are required to take 1 to 1. An element a of a ring is a unit if it has an inverse (element b such that ab 1 ba). The identity element of a ring is required to act as 1 on a module over D D the ring. X YX is a subset of Y (not necessarily proper); X def YX is defined to be Y , or equals Y by definition; D X YX is isomorphic to Y ; X YX and Y are canonically isomorphic (or there is a given or unique isomorphism); ' PREREQUISITES An undergraduate “abstract algebra” course. COMPUTER ALGEBRA PROGRAMS GAP is an open source computer algebra program, emphasizing computational group theory. To get started with GAP, I recommend going to Alexander Hulpke’s page here, where you will find versions of GAP for both Windows and Macs and a guide “Abstract Algebra in GAP”. The Sage page here provides a front end for GAP and other programs. I also recommend N. Carter’s “Group Explorer” here for exploring the structure of groups of small order. Earlier versions of these notes (v3.02) described how to use Maple for computations in group theory. ACKNOWLEDGEMENTS I thank the following for providing corrections and comments for earlier versions of these notes: V.V. Acharya; Yunghyun Ahn; Max Black; Tony Bruguier; Vigen Chaltikyan; Dustin Clausen; Benoˆıt Claudon; Keith Conrad; Demetres Christofides; Adam Glesser; Darij Grinberg; Sylvan Jacques; Martin Klazar; Thomas Lamm; Mark Meckes; Max Menzies; Victor Petrov; Flavio Poletti; Diego Silvera; Efthymios Sofos; Dave Simpson; David Speyer; Kenneth Tam; Robert Thompson; Bhupendra Nath Tiwari; Leandro Vendramin; Michiel Vermeulen. Also, I have benefited from the posts to mathoverflow by Richard Borcherds, Robin Chapman, Steve Dalton, Leonid Positselski, Noah Snyder, Richard Stanley, Qiaochu Yuan, and others. A reference monnnn means question nnnn on mathoverflow.net and sxnnnn similarly refers to math.stackexchange.com. 1 A family should be distinguished from a set. For example, if f is the function Z Z=3Z sending an ! integer to its equivalence class, then f .i/ i Z is a set with three elements whereas .f .i//i Z is family with an infinite index set. f j 2 g 2 5 The theory of groups of finite order may be said to date from the time of Cauchy. To him are due the first attempts at classification with a view to forming a theory froma number of isolated facts. Galois introduced into the theory the exceedingly important idea of a [normal] sub-group, and the corresponding division of groups into simple and composite. Moreover, by shewing that to every equation of finite degree there corresponds a group of finite order on which all the properties of the equation depend, Galois indicated how far reaching the applications of the theory might be, and thereby contributed greatly, if indirectly, to its subsequent developement. Many additions were made, mainly by French mathematicians, during the middle part of the [nineteenth] century. The first connected exposition of the theory was given in the third edition of M. Serret’s “Cours d’Algebre` Superieure,´ ” which was published in 1866. This was followed in 1870 by M. Jordan’s “Traite´ des substitutions et des equations´ algebriques.´ ” The greater part of M. Jordan’s treatise is devoted to a developement of the ideas of Galois and to their application to the theory of equations. No considerable progress in the theory, as apart from its applications, was made till the appearance in 1872 of Herr Sylow’s memoir “Theor´ emes` sur les groupes de substitutions” in the fifth volume of the Mathematische Annalen. Since the date of this memoir, but more especially in recent years, the theory has
Load more

Recommended publications
  • Math 602, Fall 2002. BIII (A). We First Prove the Following Proposition
    Homework I (due September 30), Math 602, Fall 2002. BIII (a). We first prove the following proposition: Proposition 1.1 Given a group G, for any finite normal subgroup, H, of G and any p-Sylow subgroup P of H, we have G = NG(P )H. Proof . For every g ∈ G, clearly, g−1P g is a subgroup of g−1Hg = H, since H is normal in G. Since |g−1P g| = |P |, the subgroup g−1P g is also a p-Sylow subgroup of H. By Sylow II, g−1P g is some conjugate of P in H, i.e., g−1P g = hP h−1 for some h ∈ H. −1 −1 Thus, ghP h g = P , which implies that gh ∈ NG(P ), so g ∈ NG(P )H. Since the reasoning holds for every g ∈ G, we get G = NG(P )H. Now, since H is normal in G, by the second homomorphism theorem, we know that ∼ G/H = (NG(P )H)/H = NG(P )/(NG(P ) ∩ H). Moreover, it is clear that NG(P ) ∩ H = ∼ NH (P ), so G/H = NG(P )/NH (P ), as desired. (b) We shall prove that every p-Sylow subgroup of Φ(G) is normal in Φ(G) (and in fact, in G). From this, we will deduce that Φ(G) has property (N). Indeed, if we inspect the proof of the proposition proved in class stating that if G is a finite group that has (N), then G is isomorphic to the product of its p-Sylow subgroups, we see that this proof only depends on the fact that every p-Sylow subgroup of G is normal in G.
    [Show full text]
  • Arithmetic Equivalence and Isospectrality
    ARITHMETIC EQUIVALENCE AND ISOSPECTRALITY ANDREW V.SUTHERLAND ABSTRACT. In these lecture notes we give an introduction to the theory of arithmetic equivalence, a notion originally introduced in a number theoretic setting to refer to number fields with the same zeta function. Gassmann established a direct relationship between arithmetic equivalence and a purely group theoretic notion of equivalence that has since been exploited in several other areas of mathematics, most notably in the spectral theory of Riemannian manifolds by Sunada. We will explicate these results and discuss some applications and generalizations. 1. AN INTRODUCTION TO ARITHMETIC EQUIVALENCE AND ISOSPECTRALITY Let K be a number field (a finite extension of Q), and let OK be its ring of integers (the integral closure of Z in K). The Dedekind zeta function of K is defined by the Dirichlet series X s Y s 1 ζK (s) := N(I)− = (1 N(p)− )− I OK p − ⊆ where the sum ranges over nonzero OK -ideals, the product ranges over nonzero prime ideals, and N(I) := [OK : I] is the absolute norm. For K = Q the Dedekind zeta function ζQ(s) is simply the : P s Riemann zeta function ζ(s) = n 1 n− . As with the Riemann zeta function, the Dirichlet series (and corresponding Euler product) defining≥ the Dedekind zeta function converges absolutely and uniformly to a nonzero holomorphic function on Re(s) > 1, and ζK (s) extends to a meromorphic function on C and satisfies a functional equation, as shown by Hecke [25]. The Dedekind zeta function encodes many features of the number field K: it has a simple pole at s = 1 whose residue is intimately related to several invariants of K, including its class number, and as with the Riemann zeta function, the zeros of ζK (s) are intimately related to the distribution of prime ideals in OK .
    [Show full text]
  • Affine Springer Fibers and Affine Deligne-Lusztig Varieties
    Affine Springer Fibers and Affine Deligne-Lusztig Varieties Ulrich G¨ortz Abstract. We give a survey on the notion of affine Grassmannian, on affine Springer fibers and the purity conjecture of Goresky, Kottwitz, and MacPher- son, and on affine Deligne-Lusztig varieties and results about their dimensions in the hyperspecial and Iwahori cases. Mathematics Subject Classification (2000). 22E67; 20G25; 14G35. Keywords. Affine Grassmannian; affine Springer fibers; affine Deligne-Lusztig varieties. 1. Introduction These notes are based on the lectures I gave at the Workshop on Affine Flag Man- ifolds and Principal Bundles which took place in Berlin in September 2008. There are three chapters, corresponding to the main topics of the course. The first one is the construction of the affine Grassmannian and the affine flag variety, which are the ambient spaces of the varieties considered afterwards. In the following chapter we look at affine Springer fibers. They were first investigated in 1988 by Kazhdan and Lusztig [41], and played a prominent role in the recent work about the “fun- damental lemma”, culminating in the proof of the latter by Ngˆo. See Section 3.8. Finally, we study affine Deligne-Lusztig varieties, a “σ-linear variant” of affine Springer fibers over fields of positive characteristic, σ denoting the Frobenius au- tomorphism. The term “affine Deligne-Lusztig variety” was coined by Rapoport who first considered the variety structure on these sets. The sets themselves appear implicitly already much earlier in the study of twisted orbital integrals. We remark that the term “affine” in both cases is not related to the varieties in question being affine, but rather refers to the fact that these are notions defined in the context of an affine root system.
    [Show full text]
  • Classification of Finite Abelian Groups
    Math 317 C1 John Sullivan Spring 2003 Classification of Finite Abelian Groups (Notes based on an article by Navarro in the Amer. Math. Monthly, February 2003.) The fundamental theorem of finite abelian groups expresses any such group as a product of cyclic groups: Theorem. Suppose G is a finite abelian group. Then G is (in a unique way) a direct product of cyclic groups of order pk with p prime. Our first step will be a special case of Cauchy’s Theorem, which we will prove later for arbitrary groups: whenever p |G| then G has an element of order p. Theorem (Cauchy). If G is a finite group, and p |G| is a prime, then G has an element of order p (or, equivalently, a subgroup of order p). ∼ Proof when G is abelian. First note that if |G| is prime, then G = Zp and we are done. In general, we work by induction. If G has no nontrivial proper subgroups, it must be a prime cyclic group, the case we’ve already handled. So we can suppose there is a nontrivial subgroup H smaller than G. Either p |H| or p |G/H|. In the first case, by induction, H has an element of order p which is also order p in G so we’re done. In the second case, if ∼ g + H has order p in G/H then |g + H| |g|, so hgi = Zkp for some k, and then kg ∈ G has order p. Note that we write our abelian groups additively. Definition. Given a prime p, a p-group is a group in which every element has order pk for some k.
    [Show full text]
  • Dynamics for Discrete Subgroups of Sl 2(C)
    DYNAMICS FOR DISCRETE SUBGROUPS OF SL2(C) HEE OH Dedicated to Gregory Margulis with affection and admiration Abstract. Margulis wrote in the preface of his book Discrete subgroups of semisimple Lie groups [30]: \A number of important topics have been omitted. The most significant of these is the theory of Kleinian groups and Thurston's theory of 3-dimensional manifolds: these two theories can be united under the common title Theory of discrete subgroups of SL2(C)". In this article, we will discuss a few recent advances regarding this missing topic from his book, which were influenced by his earlier works. Contents 1. Introduction 1 2. Kleinian groups 2 3. Mixing and classification of N-orbit closures 10 4. Almost all results on orbit closures 13 5. Unipotent blowup and renormalizations 18 6. Interior frames and boundary frames 25 7. Rigid acylindrical groups and circular slices of Λ 27 8. Geometrically finite acylindrical hyperbolic 3-manifolds 32 9. Unipotent flows in higher dimensional hyperbolic manifolds 35 References 44 1. Introduction A discrete subgroup of PSL2(C) is called a Kleinian group. In this article, we discuss dynamics of unipotent flows on the homogeneous space Γn PSL2(C) for a Kleinian group Γ which is not necessarily a lattice of PSL2(C). Unlike the lattice case, the geometry and topology of the associated hyperbolic 3-manifold M = ΓnH3 influence both topological and measure theoretic rigidity properties of unipotent flows. Around 1984-6, Margulis settled the Oppenheim conjecture by proving that every bounded SO(2; 1)-orbit in the space SL3(Z)n SL3(R) is compact ([28], [27]).
    [Show full text]
  • Mathematics 310 Examination 1 Answers 1. (10 Points) Let G Be A
    Mathematics 310 Examination 1 Answers 1. (10 points) Let G be a group, and let x be an element of G. Finish the following definition: The order of x is ... Answer: . the smallest positive integer n so that xn = e. 2. (10 points) State Lagrange’s Theorem. Answer: If G is a finite group, and H is a subgroup of G, then o(H)|o(G). 3. (10 points) Let ( a 0! ) H = : a, b ∈ Z, ab 6= 0 . 0 b Is H a group with the binary operation of matrix multiplication? Be sure to explain your answer fully. 2 0! 1/2 0 ! Answer: This is not a group. The inverse of the matrix is , which is not 0 2 0 1/2 in H. 4. (20 points) Suppose that G1 and G2 are groups, and φ : G1 → G2 is a homomorphism. (a) Recall that we defined φ(G1) = {φ(g1): g1 ∈ G1}. Show that φ(G1) is a subgroup of G2. −1 (b) Suppose that H2 is a subgroup of G2. Recall that we defined φ (H2) = {g1 ∈ G1 : −1 φ(g1) ∈ H2}. Prove that φ (H2) is a subgroup of G1. Answer:(a) Pick x, y ∈ φ(G1). Then we can write x = φ(a) and y = φ(b), with a, b ∈ G1. Because G1 is closed under the group operation, we know that ab ∈ G1. Because φ is a homomorphism, we know that xy = φ(a)φ(b) = φ(ab), and therefore xy ∈ φ(G1). That shows that φ(G1) is closed under the group operation.
    [Show full text]
  • LINEAR ALGEBRA METHODS in COMBINATORICS László Babai
    LINEAR ALGEBRA METHODS IN COMBINATORICS L´aszl´oBabai and P´eterFrankl Version 2.1∗ March 2020 ||||| ∗ Slight update of Version 2, 1992. ||||||||||||||||||||||| 1 c L´aszl´oBabai and P´eterFrankl. 1988, 1992, 2020. Preface Due perhaps to a recognition of the wide applicability of their elementary concepts and techniques, both combinatorics and linear algebra have gained increased representation in college mathematics curricula in recent decades. The combinatorial nature of the determinant expansion (and the related difficulty in teaching it) may hint at the plausibility of some link between the two areas. A more profound connection, the use of determinants in combinatorial enumeration goes back at least to the work of Kirchhoff in the middle of the 19th century on counting spanning trees in an electrical network. It is much less known, however, that quite apart from the theory of determinants, the elements of the theory of linear spaces has found striking applications to the theory of families of finite sets. With a mere knowledge of the concept of linear independence, unexpected connections can be made between algebra and combinatorics, thus greatly enhancing the impact of each subject on the student's perception of beauty and sense of coherence in mathematics. If these adjectives seem inflated, the reader is kindly invited to open the first chapter of the book, read the first page to the point where the first result is stated (\No more than 32 clubs can be formed in Oddtown"), and try to prove it before reading on. (The effect would, of course, be magnified if the title of this volume did not give away where to look for clues.) What we have said so far may suggest that the best place to present this material is a mathematics enhancement program for motivated high school students.
    [Show full text]
  • NOTES on FINITE GROUP REPRESENTATIONS in Fall 2020, I
    NOTES ON FINITE GROUP REPRESENTATIONS CHARLES REZK In Fall 2020, I taught an undergraduate course on abstract algebra. I chose to spend two weeks on the theory of complex representations of finite groups. I covered the basic concepts, leading to the classification of representations by characters. I also briefly addressed a few more advanced topics, notably induced representations and Frobenius divisibility. I'm making the lectures and these associated notes for this material publicly available. The material here is standard, and is mainly based on Steinberg, Representation theory of finite groups, Ch 2-4, whose notation I will mostly follow. I also used Serre, Linear representations of finite groups, Ch 1-3.1 1. Group representations Given a vector space V over a field F , we write GL(V ) for the group of bijective linear maps T : V ! V . n n When V = F we can write GLn(F ) = GL(F ), and identify the group with the group of invertible n × n matrices. A representation of a group G is a homomorphism of groups φ: G ! GL(V ) for some representation choice of vector space V . I'll usually write φg 2 GL(V ) for the value of φ on g 2 G. n When V = F , so we have a homomorphism φ: G ! GLn(F ), we call it a matrix representation. matrix representation The choice of field F matters. For now, we will look exclusively at the case of F = C, i.e., representations in complex vector spaces. Remark. Since R ⊆ C is a subfield, GLn(R) is a subgroup of GLn(C).
    [Show full text]
  • The Classification and the Conjugacy Classesof the Finite Subgroups of The
    Algebraic & Geometric Topology 8 (2008) 757–785 757 The classification and the conjugacy classes of the finite subgroups of the sphere braid groups DACIBERG LGONÇALVES JOHN GUASCHI Let n 3. We classify the finite groups which are realised as subgroups of the sphere 2 braid group Bn.S /. Such groups must be of cohomological period 2 or 4. Depend- ing on the value of n, we show that the following are the maximal finite subgroups of 2 Bn.S /: Z2.n 1/ ; the dicyclic groups of order 4n and 4.n 2/; the binary tetrahedral group T ; the binary octahedral group O ; and the binary icosahedral group I . We give geometric as well as some explicit algebraic constructions of these groups in 2 Bn.S / and determine the number of conjugacy classes of such finite subgroups. We 2 also reprove Murasugi’s classification of the torsion elements of Bn.S / and explain 2 how the finite subgroups of Bn.S / are related to this classification, as well as to the 2 lower central and derived series of Bn.S /. 20F36; 20F50, 20E45, 57M99 1 Introduction The braid groups Bn of the plane were introduced by E Artin in 1925[2;3]. Braid groups of surfaces were studied by Zariski[41]. They were later generalised by Fox to braid groups of arbitrary topological spaces via the following definition[16]. Let M be a compact, connected surface, and let n N . We denote the set of all ordered 2 n–tuples of distinct points of M , known as the n–th configuration space of M , by: Fn.M / .p1;:::; pn/ pi M and pi pj if i j : D f j 2 ¤ ¤ g Configuration spaces play an important roleˆ in several branches of mathematics and have been extensively studied; see Cohen and Gitler[9] and Fadell and Husseini[14], for example.
    [Show full text]
  • On Some Generation Methods of Finite Simple Groups
    Introduction Preliminaries Special Kind of Generation of Finite Simple Groups The Bibliography On Some Generation Methods of Finite Simple Groups Ayoub B. M. Basheer Department of Mathematical Sciences, North-West University (Mafikeng), P Bag X2046, Mmabatho 2735, South Africa Groups St Andrews 2017 in Birmingham, School of Mathematics, University of Birmingham, United Kingdom 11th of August 2017 Ayoub Basheer, North-West University, South Africa Groups St Andrews 2017 Talk in Birmingham Introduction Preliminaries Special Kind of Generation of Finite Simple Groups The Bibliography Abstract In this talk we consider some methods of generating finite simple groups with the focus on ranks of classes, (p; q; r)-generation and spread (exact) of finite simple groups. We show some examples of results that were established by the author and his supervisor, Professor J. Moori on generations of some finite simple groups. Ayoub Basheer, North-West University, South Africa Groups St Andrews 2017 Talk in Birmingham Introduction Preliminaries Special Kind of Generation of Finite Simple Groups The Bibliography Introduction Generation of finite groups by suitable subsets is of great interest and has many applications to groups and their representations. For example, Di Martino and et al. [39] established a useful connection between generation of groups by conjugate elements and the existence of elements representable by almost cyclic matrices. Their motivation was to study irreducible projective representations of the sporadic simple groups. In view of applications, it is often important to exhibit generating pairs of some special kind, such as generators carrying a geometric meaning, generators of some prescribed order, generators that offer an economical presentation of the group.
    [Show full text]
  • Discrete and Profinite Groups Acting on Regular Rooted Trees
    Discrete and Profinite Groups Acting on Regular Rooted Trees Dissertation zur Erlangung des mathematisch-naturwissenschaftlichen Doktorgrades “Doctor rerum naturalium” der Georg-August-Universit¨at G¨ottingen vorgelegt von Olivier Siegenthaler aus Lausanne G¨ottingen, den 31. August 2009 Referent: Prof. Dr. Laurent Bartholdi Korreferent: Prof. Dr. Thomas Schick Tag der m¨undlichen Pr¨ufung: den 28. September 2009 Contents Introduction 1 1 Foundations 5 1.1 Definition of Aut X∗ and ...................... 5 A∗ 1.2 ZariskiTopology ............................ 7 1.3 Actions of X∗ .............................. 8 1.4 Self-SimilarityandBranching . 9 1.5 Decompositions and Generators of Aut X∗ and ......... 11 A∗ 1.6 The Permutation Modules k X and k X ............ 13 { } {{ }} 1.7 Subgroups of Aut X∗ .......................... 14 1.8 RegularBranchGroups . 16 1.9 Self-Similarity and Branching Simultaneously . 18 1.10Questions ................................ 19 ∗ 2 The Special Case Autp X 23 2.1 Definition ................................ 23 ∗ 2.2 Subgroups of Autp X ......................... 24 2.3 NiceGeneratingSets. 26 2.4 Uniseriality ............................... 28 2.5 Signature and Maximal Subgroups . 30 2.6 Torsion-FreeGroups . 31 2.7 Some Specific Classes of Automorphisms . 32 2.8 TorsionGroups ............................. 34 3 Wreath Product of Affine Group Schemes 37 3.1 AffineSchemes ............................. 38 3.2 ExponentialObjects . 40 3.3 Some Hopf Algebra Constructions . 43 3.4 Group Schemes Corresponding to Aut X∗ .............. 45 3.5 Iterated Wreath Product of the Frobenius Kernel . 46 i Contents 4 Central Series and Automorphism Towers 49 4.1 Notation................................. 49 4.2 CentralSeries.............................. 51 4.3 Automorphism and Normalizer Towers . 54 5 Hausdorff Dimension 57 5.1 Definition ................................ 57 5.2 Layers .................................. 58 5.3 ComputingDimensions .
    [Show full text]
  • A CHARACTERISTIC SUBGROUP of a P-GROUP
    Pacific Journal of Mathematics A CHARACTERISTIC SUBGROUP OF A p-GROUP CHARLES RAY HOBBY Vol. 10, No. 3 November 1960 A CHARACTERISTIC SUBGROUP OF A p-GROVP CHARLES HOBBY If x, y are elements and H, K subsets of the p-group G, we shall denote by [x, y] the element y~px~p(xy)p of G, and by [H, K] the sub- group of G generated by the set of all [h, k] for h in H and k in K. We call a p-group G p-abelίan if (xy)p = xpyp for all elements x, y of G. If we let Θ(G) — [G, G] then #(G) is a characteristic subgroup of G and Gjθ{G) is p-abelian. In fact, Θ(G) is the minimal normal subgroup N of G for which G/AΓ is p-abelian. It is clear that Θ(G) is contained in the derived group of G, and G/Θ(G) is regular in the sense of P. Hall [3] Theorem 1 lists some elementary properties of p-abelian groups. These properties are used to obtain a characterization of p-groups G (for p > 3) in which the subgroup generated by the pth powers of elements of G coincides with the Frattini subgroup of G (Theorems 2 and 3). A group G is said to be metacyclic if there exists a cyclic normal sub- group N with G/N cyclic. Theorem 4 states that a p-group G, for p > 2, is metacyclic if and only if Gjθ(G) is metacyclic.
    [Show full text]