A HISTORY OF FINITE SIMPLE GROUPS by Faun CC Doherty BA Ob erlin College OH A thesis submitted to the University of Colorado at Denver in partial fulllment of the requirements for the degree of Master of Science Applied Mathematics This thesis for the Master of Science degree by Faun CC Doherty has b een approved by J Richard Lundgren William E Cherowitzo Stanley E Payne Date Doherty Faun CC MS Applied Mathematics A History of Finite Simple Groups Thesis directed by Asso ciate Professor J Richard Lundgren ABSTRACT A group is a set together with an asso ciative binary op eration such that there exists an identity element for the set and an inverse for each element in the set All nite groups can b e broken down into a series of nite simple groups which have b een called the building blo cks of nite groups The history of nite simple groups originates in the s with Evariste Galois and the solution of fth degree p olynomial equations In the twentieth century the recognition of the imp ortance of nite simple groups inspired a huge eort to nd all nite simple groups This classication pro ject was completed in We shall b egin by taking a historical lo ok at the earliest metho ds of analyzing the structure of nite groups according to their order Finite simple groups can b e divided into two typ es those b elonging to innite families and the sp oradic simple groups We shall lo ok at the discovery and representation of many of these Finally we shall discuss the monumental to page pro of of the classication of all nite simple groups This abstract accurately represents the content of the candidates thesis I recommend its publication Signed J Richard Lundgren iii ACKNOWLEDGEMENTS I would like to thank Professor Lundgren for his supp ort in writing this thesis Also thanks to my parents for their example and Michael for his patience CONTENTS Chapter Intro duction The Range Problem Intro duction to the Problem Sylows Theorems Other Theorems Corollaries Etc That Will Prove Useful Some History Holder Cole Burnside The Completion of the Range Problem Through Order One Million Some Examples The Simple Groups Innite Families of Simple Groups The Alternating Groups Simple Groups of Lie Typ e The Classical Linear Groups Other Lie Groups The Sp oradic Simple Groups v The Mathieu Groups Centralizer of Involution Problems Rank Permutation Groups The Remaining Sp oradic Simple Groups The Classication Theorem History The Theorem References vi Intro duction Some have referred to the study of simple groups as the El Dorado of nite group theory It has b een a very active eld of study through the twentieth century and has its ro ots in the nineteenth as do es group theory itself A group is dened as a set together with an asso ciative binary op eration dened such that there exist an identity element for the set and inverses for each element of the set The set is closed under the op eration A normal subgroup H of a group G is a subgroup such that aH H a for all a G 1 Another denition of normal is that a H a H for a G A simple group is a group which has no normal subgroups except itself and the identity which are always normal Those groups with prime order have no subgroups except for the identity and the group itself thus they are considered trivially simple For the rest of this pap er the term simple group will refer to nite nontrivial simple groups Simple groups are sp ecial kinds of groups that are the building blo cks of all other groups thus the imp ortance in their study This idea was recognized as early as by Evariste Galois and later a search for the simple groups to ok place In the twentieth century this search culminated in a monumental theorem which classies all simple groups One of the earliest metho ds of lo cating simple groups is called the range problem This is a systematic examination of the internal structure of groups according to the order of the group Chapter one of this pap er will outline the history of this problem and the metho ds used through the analysis of groups up to order one million The second chapter will describ e the simple groups by their typ es innite families of simple groups and the sp oradic simple groups How some of these groups can b e represented as well as the metho ds of their discovery will b e discussed Finally a general outline of the classication theorem will b e given in the last chapter The Range Problem Intro duction to the Problem Among the metho ds of determining all nite simple groups the ap proach of examining individual groups of certain orders can seem at times slow and metho dical Yet this task b egun in by Otto Holder has proven fruit ful in the advancement of group theory if not always in the discovery of new simple groups It has shed a great deal of light up on the structure of groups with given orders which allows one to understand the nature of simple groups at least in so far as determining what they are not This particular prob lem lasted through to when Marshall Hall Jr completed the individual examination of groups with particular orders through the order of Ab out eleven individuals from to participated in the solution of this problem each aided by the work and discoveries of those who came b efore The range problem itself is not dicult to understand in light of the search for simple groups It is simply this given a particular natural numb er say n what can we say ab out the structure of any group having n elements And in particular can we determine if the group has any normal subgroups b esides itself and the identity ie can we show that the group is not simple If the group is simple is it unique Through the history of this problem there were two main metho ds used to explore the structure of groups with a given order One was to use the Sylow theorems and the other was to employ character theory It will b e the task of this pap er to concentrate only on the Sylow theorem metho ds thus a word ab out these theorems is in order Sylows Theorems Ludvig Sylow a Norwegian mathematician came up with the Sylow theorems in by way of the study of p ermutation group theory These results lost no imp ortance with the development of abstract group theory in fact their imp ortance grew The Sylow theorems as we state and prove them to day are based on the fundamental concept known as Lagranges theorem and it is here that we shall start Theorem Lagranges theorem Suppose H G is a subgroup Then jGj jH j jG H j Note that jG H j is the index of H in G or the numb er of distinct right cosets of H in G A right coset is the set H g fhg j h H g where H G We can show easily that the group G is the disjoint union of the distinct right cosets The cardinality of each coset is equal to the numb er of elements in the subgroup H and with these two facts we may deduce that the order of G is the order of H times the numb er of distinct right cosets that partition G k Theorem Sylows Theorem If p j jGj then G has at least one k subgroup of order p for any prime p Thus if any p ower of a prime divides the order of our group then the group has a subgroup of order that p ower of the prime k Theorem Sylows Theorem If H G and jH j p then H is contained in some Sylow psubgroup A Sylow psubgroup is a subgroup of G such that its order is equal to the full p ower of p in the order of G For example if we have a group of 4 2 4 order a sylow subgroup would have order The set of all Sylow psubgroups of G is denoted S y l G We know from the rst Sylow theorem p that S y l G is not empty We can also nd a Corollary b elow which p states that if only one Sylow psubgroup exists then it is normal in G This fact will allow us to eliminate easily many integers as p ossible orders of simple groups Theorem Sylows Theorem The number of Sylow psubgroups of G ie jS y l Gjwritten n has the fol lowing properties p p n mod p p e and n mod p p e if p jS S T j for al l S and T S y l G with S T p The examination of the structure of groups with a given order is feasible b e cause of a numb er of other results b esides the Sylow theorems although many of these results are based on the Sylow theorems A numb er of these results shall b e listed b elow and referred to throughout this chapter Other Theorems Corollaries Etc That Will Prove Useful Theorem A nontrivial nite pgroup has a nontrivial center A pgroup where p is prime is dened as a group in which every element has order a p ower of p The center of a group Z G is a normal subgroup of G comp osed of all elements which commute with all other elements of G a Theorem If jGj p where p is prime and a then G is not simple a Pro of Let jGj p and supp ose that G is simple Since G itself is a pgroup by we know that Z G G and since G is simple Z G must b e G But then G is ab elian and its simplicity implies that jGj p This is a a contradiction since jGj p so G is not simple Theorem The NC Theorem If H G then the factor group of the normalizer of H in G by the centralizer of H in G is isomorphic to