San Jose State University SJSU ScholarWorks Master's Theses Master's Theses and Graduate Research Summer 2011 Construction and Simplicity of the Large Mathieu Groups Robert Peter Hansen San Jose State University Follow this and additional works at: https://scholarworks.sjsu.edu/etd_theses Recommended Citation Hansen, Robert Peter, "Construction and Simplicity of the Large Mathieu Groups" (2011). Master's Theses. 4053. DOI: https://doi.org/10.31979/etd.qnhv-a5us https://scholarworks.sjsu.edu/etd_theses/4053 This Thesis is brought to you for free and open access by the Master's Theses and Graduate Research at SJSU ScholarWorks. It has been accepted for inclusion in Master's Theses by an authorized administrator of SJSU ScholarWorks. For more information, please contact [email protected]. CONSTRUCTION AND SIMPLICITY OF THE LARGE MATHIEU GROUPS A Thesis Presented to The Faculty of the Department of Mathematics San Jos´eState University In Partial Fulfillment of the Requirements for the Degree Master of Science by R. Peter Hansen August 2011 c 2011 R. Peter Hansen ALL RIGHTS RESERVED The Designated Thesis Committee Approves the Thesis Titled CONSTRUCTION AND SIMPLICITY OF THE LARGE MATHIEU GROUPS by R. Peter Hansen APPROVED FOR THE DEPARTMENT OF MATHEMATICS SAN JOSE´ STATE UNIVERSITY August 2011 Dr. Timothy Hsu Department of Mathematics Dr. Roger Alperin Department of Mathematics Dr. Brian Peterson Department of Mathematics ABSTRACT CONSTRUCTION AND SIMPLICITY OF THE LARGE MATHIEU GROUPS by R. Peter Hansen In this thesis, we describe the construction of the Mathieu group M24 given by Ernst Witt in 1938, a construction whose geometry was examined by Jacques Tits in 1964. This construction is achieved by extending the projective semilinear group 2 2 P ΓL3(F4) and its action on the projective plane P (F4). P (F4) is the projective plane over the field of 4 elements, with 21 points and 21 lines, and P ΓL3(F4) is the 2 largest group sending lines to lines in P (F4). This plane has 168 6-point subsets, hexads, with the property that no 3 points of a hexad are collinear. Under the 2 action of the subgroup P SL3(F4), the hexads in P (F4) break into 3 orbits of equal size. These orbits are preserved and permuted by P ΓL3(F4), and can be viewed as 3 2 points, which, when added to the 21 points of P (F4), yield a set X of 24 points. 2 Using lines and hexads in P (F4), we define certain 8-point subsets of X, view them 24 as vectors in F2 , and define the subspace they span as the Golay 24-code. We then define M24 as the automorphism group of the Golay 24-code and show that it acts 5-transitively on X, establishing its simplicity. We calculate the order of M24 and the order of two simple subgroups, M23 and M22, the other large Mathieu groups. ACKNOWLEDGEMENTS I want to thank my wife, Claire. Without her constant encouragement, I could not have finished a master's degree in mathematics. She has my enduring gratitude. I would like to thank my thesis advisor, Tim Hsu. I would have been quickly lost in navigating the \Witt-Tits" construction had it not been for his guidance. I also would like to thank Brian Peterson and Roger Alperin for their courses in abstract algebra, where my interest in group theory was first sparked. v TABLE OF CONTENTS CHAPTER 1 INTRODUCTION 1 2 PREPARATORY LEMMAS 5 2.1 Normal and Characteristic Subgroups . 5 2.2 Cosets and Products of Groups . 10 2.3 p-Groups . 13 2.4 Cyclic Groups . 16 2.5 Field Theory Lemmas . 21 2.6 Vector Space Lemmas . 23 3 GROUP ACTIONS AND MULTIPLE TRANSITIVITY 29 3.1 Group Actions . 29 3.2 Transitivity, Orbits, and Stabilizers . 30 3.3 Multiple Transitivity and Sharp Transitivity . 39 3.4 Primitive Actions . 44 3.5 Simplicity of Multiply-Transitive Groups . 45 4 LINEAR GROUPS AND THE SIMPLICITY OF P SL3(F4) 51 4.1 Linear Groups . 51 4.2 Action of Linear Groups on Projective Space . 57 4.3 Simplicity of P SLn(F ).......................... 62 vi 5 SEMILINEAR GROUPS 66 5.1 Field Extensions and Automorphism Groups . 66 5.2 Semilinear Groups . 71 5.3 Action of Semilinear Groups on Projective Space . 80 6 BILINEAR FORMS 84 6.1 Bilinear Forms . 84 6.2 Congruence Classes . 88 6.3 Symmetric and Alternate Forms . 94 6.4 Reflexive Forms and Dual Spaces . 96 6.5 Orthogonal Complements . 101 6.6 Classification of Alternate Forms . 106 7 QUADRATIC FORMS IN CHARACTERISTIC 2 109 7.1 Quadratic Forms . 109 7.2 Quadratic Forms and Homogeneous Polynomials . 112 7.3 Congruence Classes . 115 7.4 Classification of Regular Quadratic Forms . 118 8 CURVES IN THE PROJECTIVE PLANE 128 8.1 Projective Space . 128 8.2 Action of ΓLn(E) on Zero Sets of Homogeneous Polynomials . 131 8.3 Duality . 137 8.4 Lines . 140 8.5 Regular Conics and Hyperconics . 142 2 9 HEXADS IN P (F4) 147 2 9.1 k-Arcs in P (Fq) ............................. 147 vii 9.2 Orbits of Hexads . 151 9.3 Hexagrams . 155 9.4 Even Intersections and Hexad Orbits . 162 10 BINARY LINEAR CODES 175 10.1 Binary Linear Codes . 175 10.2 Self-Orthogonal and Self-Dual Codes . 177 10.3 Automorphism Group of a Binary Code . 182 11 LARGE MATHIEU GROUPS 185 11.1 Action of P ΓL3(F4) on P SL3(F4)-Orbits of Hexads . 185 11.2 Golay Code C24 .............................. 188 11.3 Large Mathieu Groups . 191 11.4 Steiner System of Octads . 196 11.5 Simplicity of the Large Mathieu Groups . 198 BIBLIOGRAPHY 204 viii CHAPTER 1 INTRODUCTION A simple group is one whose normal subgroups are trivial. Normal subgroups allow the definition of factor groups, a decomposition of a group. Having no such decomposition, simple groups are \atoms of symmetry" (Ronan [Ron06]). It was Evariste Galois who first noted the importance of normal subgroups in 1832, but his description remained unpublished until 1846 (Galois [Gal46]). In 1870, Camille Jordan, along with Otto Holder, determined each group has a unique decomposition as a sequence of factor groups, a \composition series" that is unique up to order and isomorphism (Jordan [Jor70]). Groups were initially conceived of as collections of permutations possessing an intrinsic coherence. The elements permuted were variously called \symbols," \letters," and \points." It was noted that some groups could permute subsets of a given size to any other subset of that size, a feature known as transitivity. If k points can be moved to any other k points, that group is k-transitive. In 1861 and 1873, a French mathematician, Emile Mathieu, published papers that described two 5-transitive groups, one acting on 12 points, and the other on 24 points, neither of which were alternating (Mathieu [Mat61], [Mat73]). The one acting on 24 points, M24, is the topic of this thesis. These groups, along with three others that are subgroups of the other two, were the first \sporadic" simple groups to be discovered. The next sporadic group was not discovered for another hundred years, in 1965 (Janko [Jan66]). Its discovery, and the proof of the odd order theorem in 1962 (Feit [FT63]), initiated a great deal of work in finite group theory, culminating with the 2 construction of the largest sporadic group, the Monster, in 1982 (Griess [Gri82]). The classification theorem for finite simple groups, mostly finished by 1983, but under revision even today, completely classifies these groups (Aschbacher [Asc04]). There are 18 countably infinite families, as well as 26 groups that do not fit into any such family (Mazurov [Maz]). These 26 are the sporadic groups, and although not part of an infinite family, they are interrelated, as all but six are subgroups or subquotients of the Monster [Ron06]. The 18 infinite families include the cyclic groups of prime order, and the alternating groups acting on 5 or more points. The next 16 families are the groups of Lie type, including the projective linear groups. One of these, P SL3(F4), is used in this thesis. It is a subgroup of M24, and plays a key role in proving its simplicity. This projective group is a natural subgroup of the projective semilinear group, P ΓL3(F4), and this larger group will be used to construct M24. There are several constructions of M24, and in fact, Mathieu did not fully convince the mathematical community he had constructed two 5-transitive groups. It was not until 1935 that Ernst Witt gave a definitive construction of M24 (Witt [Wit38a]). In 1964, Jacques Tits published a paper that explored the geometry of Witt's construction (Tits [Tit64]). Conway and Sloane refer to this construction as the \Witt-Tits" construction, and it is the one explained in this thesis (Conway and Sloane [CS93]). Witt describes a certain combinatorial structure: 759 blocks, each an 8-element subset of 24 points X, possessing the property that any 5-element subset of X lies in exactly one block. Witt showed M24 could be realized as the automorphism group of this \Steiner system" (Witt [Wit38b]). This Steiner system has a fascinating connection with modern communications. Each of the 759 blocks can be viewed as a 24-tuple over the field of two elements, and the subspace generated by these vectors is known as the Golay 3 24-code (Pless [Ple89]). It is a 12-dimensional subspace which allows 12 bits of \information" to be transmitted, along with 12 bits of \redundancy." These codewords have the property that, upon reception, any 3-bit corruption can be detected and corrected. The Mathieu group M24 can be defined as the automorphism group of the Golay 24-code.