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The University of Chicago Lazard Correspondence Up THE UNIVERSITY OF CHICAGO LAZARD CORRESPONDENCE UP TO ISOCLINISM A DISSERTATION SUBMITTED TO THE FACULTY OF THE DIVISION OF THE PHYSICAL SCIENCES IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS BY VIPUL NAIK CHICAGO, ILLINOIS DECEMBER 2013 TABLE OF CONTENTS LIST OF TABLES . xii ABSTRACT . xiii ACKNOWLEDGMENTS . xiv 1 INTRODUCTION, OUTLINE, AND PRELIMINARIES . 1 1.0.1 Background assumed . 1 1.0.2 Group and subgroup notation . 1 1.0.3 Lie ring and subring notation . 3 1.0.4 Other conventions . 4 1.1 Introduction . 4 1.1.1 The difference in tractability between groups and abelian groups . 4 1.1.2 Nilpotent groups and their relation with abelian groups . 5 1.1.3 The Lie correspondence: general remarks . 7 1.1.4 The Lie algebra for the general linear group . 8 1.1.5 The nilpotent case of the Lie correspondence . 9 1.1.6 The example of the unitriangular matrix group . 10 1.1.7 The Malcev correspondence and Lazard correspondence . 11 1.1.8 Our generalization of the Lazard correspondence . 12 1.1.9 Similarities and differences between groups and Lie rings . 12 1.1.10 Our central tool: Schur multipliers . 13 1.1.11 Globally and locally nilpotent . 14 1.1.12 The structure of this document . 15 1.1.13 For a quick reading . 17 1.2 Outline of our main results . 21 1.2.1 The Lazard correspondence: a rapid review . 21 1.2.2 Isoclinism: a rapid review . 22 1.2.3 The Lazard correspondence up to isoclinism . 23 1.2.4 The existence question . 24 1.2.5 The realization of isoclinism types . 25 1.2.6 Powering assumptions . 26 1.2.7 Global Lazard correspondence preserves Schur multipliers . 26 1.3 The abelian Lie correspondence . 27 1.3.1 Abelian groups correspond to abelian Lie rings . 27 1.3.2 Preservation of homomorphisms: viewing exp and log as functors . 28 1.3.3 Isomorphism over Set . 29 1.3.4 Equality of endomorphism monoids and of automorphism groups . 30 1.3.5 The correspondence up to isomorphism . 30 1.3.6 Isomorphism of categories versus equivalence of categories . 31 1.3.7 Subgroups, quotients, and direct products . 32 1.3.8 Characteristic and fully invariant . 34 1.3.9 How the template will be reused . 34 ii 2 ISOCLINISM AND HOMOCLINISM: BASIC THEORY . 36 2.1 Isoclinism and homoclinism for groups . 36 2.1.1 Isoclinism of groups: definition . 36 2.1.2 Homoclinism of groups . 38 2.1.3 Composition of homoclinisms . 39 2.1.4 Category of groups with homoclinisms . 39 2.1.5 Homomorphisms and homoclinisms . 40 2.1.6 Miscellaneous results on homoclinisms and words . 41 2.1.7 Isoclinic groups: how similar are they? . 44 2.1.8 Isoclinism defines a correspondence between some subgroups . 47 2.1.9 Characteristic subgroups, quotient groups, and subquotients deter- mined by the group up to isoclinism . 48 2.1.10 Correspondence between abelian subgroups . 49 2.1.11 Constructing isoclinic groups . 50 2.1.12 Hall’s purpose in introducing isoclinism . 51 2.1.13 Stem groups for a given equivalence class under isoclinism . 53 2.1.14 Some low order classification information . 54 2.2 Isoclinism and homoclinism for Lie rings . 55 2.2.1 Definitions of homoclinism and isoclinism . 55 2.2.2 Homomorphisms and homoclinisms . 58 2.2.3 Miscellaneous results on homoclinisms and words . 59 2.2.4 Isoclinic Lie rings: how similar are they? . 62 2.2.5 Isoclinism defines a correspondence between some subrings . 64 2.2.6 Characteristic subrings, quotient rings, and subquotients determined by the Lie ring up to isoclinism . 65 2.2.7 Correspondence between abelian subrings . 66 2.2.8 Constructing isoclinic Lie rings . 66 3 EXTENSION THEORY . 68 3.1 Short exact sequences and central group extensions . 68 3.1.1 Definition of short exact sequence and group extension . 68 3.1.2 Group extensions with fixed base and quotient; congruence and pseudo- congruence . 70 3.1.3 Abelian normal subgroups . 71 3.1.4 Central extensions and stem extensions . 71 3.1.5 The use of cohomology groups to classify central extensions . 72 3.1.6 Homomorphism of central extensions . 73 3.2 Short exact sequences and central extensions of Lie rings . 76 3.2.1 Definition of short exact sequence and Lie ring extension . 76 3.2.2 Lie ring extensions with fixed base and quotient; congruence and pseudo- congruence . 78 3.2.3 Central extensions and stem extensions . 79 3.2.4 The use of cohomology groups to classify central extensions . 80 3.2.5 Homomorphism of central extensions . 81 3.3 Explicit description of second cohomology group using the bar resolution . 84 iii 3.3.1 Explicit description of second cohomology group using cocycles and coboundaries . 84 3.3.2 Functoriality and automorphism group action . 85 3.3.3 Identifying cohomology classes with congruence classes of central ex- tensions . 86 3.3.4 Short exact sequence of coboundaries, cocycles, and cohomology classes 88 3.4 Exterior square, Schur multiplier, and homoclinism . 88 3.4.1 Exterior square . 89 3.4.2 The existence of a single central extension that realizes the exterior square . 92 3.4.3 Homoclinism of central extensions . 93 3.4.4 Relation between the category of central extensions and the category of central extensions with homoclinisms . 94 3.4.5 Uniqueness of homoclinism if it exists . 95 3.4.6 Existence and description of initial objects in the category of central extensions up to homoclinisms . 96 3.4.7 An alternate construction of the initial object . 98 3.4.8 Functoriality of exterior square and Schur multiplier . 100 3.4.9 Homoclinisms and words for central extensions . 100 3.5 Exterior square, Schur multiplier, and homoclinism for Lie rings . 102 3.5.1 Exterior square . 102 3.5.2 Free Lie ring on an abelian group . 104 3.5.3 Relation between exterior square of a Lie ring and exterior square in the abelian group sense . 106 3.5.4 The existence of a single central extension that realizes the exterior square . 107 3.5.5 Homoclinism of central extensions . 108 3.5.6 Relation between the category of central extensions and the category of central extensions with homoclinisms . 109 3.5.7 Uniqueness of homoclinism if it exists . 110 3.5.8 Existence and description of initial objects in the category of central extensions up to homoclinisms . 111 3.5.9 An alternate construction of the initial object . 112 3.5.10 Functoriality of exterior square and Schur multiplier . 114 3.5.11 Homoclinisms and words for central extensions . 114 3.6 Exterior square, Schur multiplier, and the second cohomology group . 116 3.6.1 Homomorphism from the Schur multiplier to the kernel of the extension117 3.6.2 Classification of extensions up to isoclinism . 118 3.6.3 Relating the classification of extensions up to isoclinism with the ho- momorphism from the Schur multiplier . 118 3.6.4 The universal coefficient theorem short exact sequence . 120 3.6.5 An alternate characterization of initial objects, and the existence of Schur covering groups . 123 3.6.6 Realizability of surjective homomorphisms from the exterior square . 126 3.6.7 The existence of stem groups . 127 iv 3.6.8 The Stallings exact sequence . 130 3.6.9 Hopf’s formula: two proofs . 131 3.6.10 Hopf’s formula: class one more version . 132 3.6.11 Exterior square of an abelian group . 132 3.6.12 Relationship with K-theory . 133 3.7 Exterior square, Schur multiplier, and the second cohomology group: Lie ring version . 134 3.7.1 Homomorphism from the Schur multiplier to the kernel of the extension134 3.7.2 Classification of extensions up to isoclinism . 135 3.7.3 Relating the classification of extensions up to isoclinism with the ho- momorphism from the Schur multiplier . 136 3.7.4 The universal coefficient theorem short exact sequence . 138 3.7.5 An alternate characterization of initial objects, and the existence of Schur covering Lie rings . 141 3.7.6 Realizability of surjective homomorphisms from the exterior square . 144 3.7.7 The existence of stem Lie rings . 145 3.7.8 The Stallings exact sequence . 147 3.7.9 Hopf’s formula for Lie rings: two proofs . 148 3.8 Exterior and tensor product for groups: explicit descriptions . 149 3.8.1 Compatible pair of actions . 150 3.8.2 Tensor product for a compatible pair of actions . 151 3.8.3 Exterior product of normal subgroups of a group . 152 3.8.4 Tensor square and exterior square of a group . 152 3.8.5 Reconciling the definitions of exterior square . 153 3.8.6 The special case of abelian groups . ..
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