Provided by the author(s) and NUI Galway in accordance with publisher policies. Please cite the published version when available. Title Classification Problems for Finite Linear Groups Author(s) Hurley, Barry Publication Date 2013-03-08 Item record http://hdl.handle.net/10379/3498 Downloaded 2021-09-27T00:50:26Z Some rights reserved. For more information, please see the item record link above. — PhD Thesis — Classification Problems for Finite Linear Groups Barry Hurley Ollscoil na hÉireann, Gaillimh National University Of Ireland, Galway Supervised by: Dr. Dane Flannery Eanáir 2012 Coimriú Is é an chloch is mó ar ár bpaidrín sa tráchtas seo ná grúpaí de chéim phríomha, agus grúpaí a + atá sainmhínithe thar réimsí críochta. Má tá q = p do p príomha agus a ∈ Z , tógaimid liosta d’fhoghrúpaí dolaghdaithe dothuaslagtha de GL(3, q) inar fíor an méid seo a leanas: • tá gach grúpa tugtha ag tacar giniúna paraiméadraithe de mhaitrísí • níl grúpaí difriúla ar an liosta comhchuingeach in GL(3, q), agus • foghrúpa dothuaslagtha dolaghdaithe ar bith de GL(3, q), tá sé GL(3, q)-chomhchuingeach le grúpa (amháin) ar an liosta. Ina theannta sin, cuirimid i gcás topaicí gaolmhara ilghnéitheacha, ina measc liosta a dhéanamh de na foghrúpaí críochta de GL(n, F) do réimsí éagsúla, F, ar sainuimhir dóibh nialas agus n ≤ 3. Úsáidimid an pacáiste ríomhaireachta Magma [4] chun ár gcuid rangui- the a chur i bhfeidhm. Pointí tagartha bunriachtanacha atá san obair a dhéantar in [2], [5] agus [11]. Contents 1 Introduction 4 1.1 Outline . .4 1.2 General approach . .5 1.3 Implementation . .7 2 Preliminaries 8 2.1 Group extensions and cohomology . .8 2.2 Some linear group theory . 12 2.2.1 Basic terms . 12 2.2.2 Actions on the underlying space . 12 2.3 Fundamentals for the classification of linear groups . 16 2.4 Linear group classification . 17 2.4.1 The isomorphism question . 19 2.4.2 The SL-transfer . 20 2.5 Changing the field . 21 2.6 Linear groups of prime degree . 22 2.7 Congruence homomorphisms . 26 3 Finite linear groups of degree two 28 4 Order conditions for finite matrix groups 35 4.1 Introduction to Blichfeldt-Dixon methods . 35 4.2 The Blichfeldt-Dixon criterion for normality of Sylow subgroups . 39 4.3 Existence of elements of composite order in finite linear groups . 44 5 Insoluble non-modular linear groups of degree three 48 5.1 Bounding group orders in degree 3 . 49 1 CONTENTS 5.2 Putting it all together: the final list . 54 5.2.1 Abstract structure of the groups . 55 5.2.2 Solution of the conjugacy problem . 56 5.2.3 Generating sets of matrices . 58 5.2.4 Some elementary number theory . 61 5.2.5 The list over finite fields . 63 6 Insoluble modular linear groups of degree three 66 6.1 Notation . 66 6.1.1 Linear group notational differences . 67 6.2 Preliminary results . 67 6.2.1 Normality of linear groups in extension groups . 69 6.3 The modular lists for SL(3, pa) and GL(3, pa) ................. 71 6.3.1 Exceptional characteristics . 71 6.3.2 Subgroups of PSL(3, pa) ......................... 72 6.3.3 Subgroups of SL(3, pa) .......................... 73 6.3.4 Subgroups of GL(3, pa) ......................... 79 6.3.5 Generating sets . 85 7 Finite linear groups over integral domains of characteristic zero 94 7.1 Preliminaries . 95 7.2 Subgroups of the general linear group of degrees 2 and 3 over a real field . 96 7.2.1 Subgroups of GL(2, K) .......................... 96 7.2.2 Subgroups of GL(3, K) .......................... 98 7.2.3 Subgroups of GL(m, Z) ......................... 101 7.3 Subgroups of degree 2 and 3 over a real number field . 103 7.3.1 Requirements on roots of polynomials for existence of subgroups . 103 7.3.2 Relating extensions by roots of unity and surds . 104 7.3.3 Lists of subgroups over extensions . 107 8 Computer implementation 111 8.1 Technical details . 111 8.1.1 Primary function: getMatrixSubgroupsOverSmallDegree ...... 112 8.1.2 Other functions . 113 8.2 Veracity and results of code . 119 8.2.1 Confirming veracity of the implemented lists . 119 2 CONTENTS 8.2.2 Table of insoluble irreducible subgroups of GL(3, q) .......... 120 8.2.3 Sample results . 120 A Notation and terminology 124 A.1 Notation . 124 A.2 Terminology . 127 Acknowledgements 130 Bibliography 131 3 Chapter 1 Introduction This thesis is concerned with classification problems for finite linear groups. Our main concern is with groups of prime degree, and groups defined over finite fields and fields of zero characteristic. We provide a glossary of basic notation and terminology in Appendix A. In this thesis, omitted proofs are either well-known, or considered to be straightforward. 1.1 Outline A major objective of the thesis is to classify completely, irredundantly, and explicitly, the insoluble irreducible subgroups of GL(3, q), for a prime power q. In other words, we will construct a list of irreducible insoluble subgroups of GL(3, q) in which • each group is given by a parametrised generating set of matrices, • different groups in the list are not conjugate in GL(3, q), and • any insoluble irreducible subgroup of GL(3, q) is GL(3, q)-conjugate to a (single) group in the list. This will also include a solution of the listing problem for the finite insoluble irreducible subgroups of GL(3, F) where F is any field of positive characteristic. The computational algebra system Magma [4] has facilities for listing the irreducible subgroups of GL(2, q), and the irreducible soluble subgroups of GL(3, q); this is work by Flannery and O’Brien [11]. The results of this thesis complete the classification of irre- ducible subgroups of GL(n, q) for n < 4 and almost all characteristics. 4 1.2. GENERAL APPROACH We also consider miscellaneous related topics: listings of the finite subgroups of GL(n, F) for various fields F of characteristic zero and n ≤ 3, and listing finite irreducible soluble subgroups of GL(n, F) for prime n and real fields F. As outlined above, by ‘classification’ we refer to the most exhaustive kind of classification that one could expect regarding linear groups – namely, by linear isomorphism (conjugacy in the ambient general linear group). While many classifications of linear groups have been obtained over the years, few are of this exhaustive type. The amount of work involved in extracting this sort of information from known lists of groups in the relevant projective special or general linear group is considerable. We begin by providing some preliminary background material, which includes basic cohomology and linear group concepts. The next chapter gives an outline of the finite linear groups of degree 2. Some of the methods used recur in the degree 3 case; however, these cases are very different. Our solution for the degree 3 case is broken up into two parts: the non-modular groups and the modular groups. We start the degree 3 case by recalling methods in that degree over the field C introduced by Blichfeldt [2, 25]; these were (much later) generalised by Dixon [5]. We combine their arguments to give a fully modernised, self-contained and complete account of the finite irreducible insoluble linear groups of degree 3 over a non-modular field. Along the way, we obtain results applicable to other degrees and fields. In Chapter 6, we complete the listing of groups in the modular case and give the corresponding matrix generating sets. Here we recall and expand on results of Bloom [3], Feit [7] and Hartley [14]. In the penultimate chapter, we present a solution of the listing problem for the finite groups of degrees 2 and 3 over certain characteristic zero fields fields and rings of algebraic integers. This chapter applies some results and general methods of the previous chapters. 1.2 General approach Let G be an irreducible subgroup of GL(n, q). Then G is completely reducible over the algebraic closure Fq of GF(q), with Fq-irreducible parts of the same degree. Hence if n is prime then G is either abelian or absolutely irreducible. The insoluble irreducible subgroups of GL(3, q) are therefore absolutely irreducible. Furthermore, such a group must be primitive over GF(q) and Fq, because otherwise it would be monomial (again using that the degree is prime), and a monomial linear group of degree less than 5 is soluble. We begin by solving the listing problem over Fq. The list over GF(q) is then obtained by determining (using trace calculations) the groups over Fq that have conjugates in GL(n, q), 5 1.2. GENERAL APPROACH and then rewriting those over GF(q). In practice, it is possible to carry out the latter task using the algorithm of Glasby and Howlett [12] for writing a finite absolutely irreducible group over a minimal field. The classification of finite linear groups is a classical problem in group theory and early results go back to the very beginning of the subject. Much of the literature here deals with the groups over the complex field C. To avail of the classical results in our attempt to list groups over finite fields, we can use a ‘reduction mod p’ homomorphism (also known as a congruence homomorphism, or Minkowski homomorphism) to ‘lift’ groups from positive to zero characteristic. The next two theorems and discussion give a flavour of the basic ideas. Theorem 1.2.1. ([5], 3.4B, p.54) Let G be a finite irreducible subgroup of GL(n, C).