On the Classification of Finite Simple Groups Niclas Bernhoff Division for Engineering Sciences, Physics and Mathemathics Karlstad University April6,2004 Abstract This paper is a small note on the classification of finite simple groups for the course ”Symmetries, Groups and Algebras” given at the Depart- ment of Physics at Karlstad University in the Spring 2004. 1Introduction In 1892 Otto Hölder began an article in Mathematische Annalen with the sen- tence ”It would be of the greatest interest if it were possible to give an overview of the entire collection of finite simple groups” (translation from German). This can be seen as the starting point of the classification of simple finite groups. In 1980 the following classification theorem could be claimed. Theorem 1 (The Classification theorem) Let G be a finite simple group. Then G is either (a) a cyclic group of prime order; (b) an alternating group of degree n 5; (c) a finite simple group of Lie type;≥ or (d) one of 26 sporadic finite simple groups. However, the work comprises in total about 10000-15000 pages in around 500 journal articles by some 100 authors. This opens for the question of possible mistakes. In fact, around 1989 Aschbacher noticed that an 800-page manuscript on quasithin groups by Mason was incomplete in various ways; especially it lacked a treatment of certain ”small” cases. Together with S. Smith, Aschbacher is still working to finish this ”last” part of the proof. Solomon predicted this to be finished in 2001-2002, but still until today the paper is not completely finished (see the homepage of S. Smith: http://www2.math.uic.edu/~smiths/). 2 Pre-classification events Even if 1892 can be seen as the year of birth of the program to classify simple finite groups, there were some important events happening before. Already 1 in 1831 normal subgroups were introduced by Evariste Galois. Galois used the concept as a tool for deciding whether a given equation was solvable by radicals. Themoremoderndefinition is as follows. Definition 1 AsubgroupH of a group G is normal if gH = Hg for all g G. ∈ Galois’ life is a tragic story. It finished in a duel 1832, when he was only 21 years old. Even if Galois was a great mathematician, reading materials for professional mathematicians at the age of fifteen, his mathematical abilities were not recognized during his lifetime. One reason was that he was not especially systematic in his work. Galois was rejected by École Polytechnique and the manuscripts he wrote for the Academy of Sciences were either rejected or got lost. First after his death Galois’ work obtained the recognition that it deserved. Camille Jordan was one of the mathematicians making comments (in 1865 and 1869) on the work of Galois. Jordan gave also the first definition of a simple group. Definition 2 AgroupG is simple if it has no proper non-trivial normal sub- groups. That is, the only normal subgroups of G are the trivial subgroup e and G itself. { } Furthermore, in a work 1870, Jordan began to build a database of finite simple groups, including the alternating groups of degree at least 5 and most of the classical projective linear groups over fields of prime cardinality. In the 1860’s Mathieu discovered 5 finite simple groups. This was the first of what later should be called the sporadic finite simple groups. Finally in 1872, the Norwegian mathematician Sylow came with his famous theorems. Sylow was born in the same year 1832 as Galois died, and spent most of his professional life as a high school teacher (he was only appointed at Chris- tiania University in 1898). Sylow stated his theorems in terms of permutation groups (the abstract definition of a group was not given yet). In 1887 the the- orems were re-proved for abstracts groups by Georg Frobenius. The theorems can be found below. Definition 3 Let p be a prime number. A group G is a p - group if every element in G has order pα for some α 0.Ap - subgroup of a group G is a subgroup of G that is a p -group. ≥ Definition 4 ASylowp - subgroup of a group G is a maximal p -subgroupof G, that is a p - subgroup contained in no larger p -subgroup. Theorem 2 (First Sylow theorem) Let G be a finite group, with order G = | | pnm,wheren 1, p is a prime number and p - m.Then 1) G containsasubgroupoforder≥ pi for 1 i n. 2) Every subgroup of order pi is a normal≤ subgroup≤ of a subgroup of order pi+1 for 1 i n 1. ≤ ≤ − 2 Theorem 3 (Second Sylow theorem) Any two Sylow p - subgroups P1 and P2 1 of a finite group G are conjugate subgroups of G,thatis,P1 = g− P2g for some g G. ∈ Theorem 4 (Third Sylow theorem) Let G be a finite group and p aprimenum- ber. If p G , then the number of Sylow p - subgroups of G is congruent to 1 modulo p ||and| divides G . | | We now apply Sylow’s theorem for some example. Example 1 No group of order 15 is simple. Let G have order 15. By Sylow’s third theorem the number of Sylow 5 - subgroupsiscongruentto1 modulo 5 and divides 15 andsothereisexactly 1 one Sylow 5 -subgroupH. The conjugates gPg− , g G,arealsoSylow5 - 1 ∈ subgroups and hence, gPg− = P for all g G. Thus there is a normal 5 - subgroup of G and so G is not simple. ∈ Remark 1 Let G be of order G = pkm,wherep is a prime and p - m.Then (in the same way as in the previous| | example) we can see that if there is a unique Sylow p - subgroup dividing G ,thenG is not simple. | | 3 Early classification results Hölder’s challenge 1892 got immediate results. Already the same year Frank Cole determined all simple groups of orders up to 500 (except some uncertainties for 360 and 432). In 1893 Cole had extended this up to the number 660, and in year 1900 Miller and Ling had reached up to 2001. The tools that they used were Sylow’s theorems and the Pigeonhole principle. The Pigeonhole Principle. If there are more than k timesasmany pigeons as pigeonholes, then some pigeonhole must contain at least k+1 pigeons. Of course, more powerful tools and new strategies were needed (the strategy of looking at one integer at a time must be changed). Some early results are: 1892 Hölder proved that a simple group whose order is a product of at • mostthreeprimefactorsmustbecyclicofprimeorder. 1893 Frobenius proved that a simple group of square-free order must be • cyclic of prime order. 1895 Both Burnside and Frobenius established that the only non-abelian • simple groups whose order is a product of at most five prime factors are PSL2(p) for p 5, 7, 11, 13 . ∈ { } 1895 Burnside proved that if G is simple, p is the smallest prime divisor • of G and G has a cyclic Sylow p - subgroup, then G = p (thisisapart of| a more| general theorem). | | 3 1900 Burnside proved that if G is a non-abelian simple group of odd order, • then G must be a product of at least seven prime numbers. Furthermore, G >| 40000| and G can have no proper subgroup H such that G / H < 101| | . | | | | 1900 L.E. Dickson extended Jordan’s database of simple groups to include • all the classical projective groups over finite fields. In the beginning of the 1900’s Dickson was constructing analogues of the Lie group G2 over finite fields F andprovedthemtobesimpleif F > 2.Healso | | studied finite analogues of E6, but didn’t prove their simplicity. Dickson’s work was completed by Chevalley and Steinberg in the 1950’s. Definition 5 AgroupG is solvable if there exists a finite sequence e = H0 { } ⊂ H1 ... Hn = G of subgroups of G,suchthatHi isanormalsubgroupofHi+1 ⊂ ⊂ for i =1, 2, ..., n 1 andsuchthatthefactorgroupsHi+1/Hi, i =1, 2, ..., n 1, are simple and abelian.− − If G is a solvable simple group, then we must have n =1in the definition and so G = G/ e is abelian. The only abelian simple groups are the cyclic groups of prime{ order.} 1904 Burnside proved the following theorem. Theorem 5 (Burnside’s paqb theorem) Let G be a finite group, such that its order has (at most) two different prime factors (i.e. G = paqb for some prime numbers p and q). Then G is a solvable group. | | In 1928 Philip Hall established generalizations of Sylow’s theorems for finite solvable groups and in 1937 he presented the following result. Theorem 6 (P. Halls theorem) G is a finite solvable group if and only if, for each expression G = mn,withm and n relatively prime, G contains at least one subgroup of| order| m and at least one subgroup of order n. From later results (before the Classification theorem) we mention just one. Nevertheless, this is a result of great importance for the Classification process. In 1960-61 a Group Theory Year was organized at the University of Chicago. This was a successful year and not least Feit and Thompson completed most of the Odd order paper - a manuscript of 255 pages, that was published in 1963 Here the authors proved the Odd order conjecture by Burnside and Miller. Theorem 7 (Odd order theorem) All finite groups of odd order are solvable. Naturally, after the Odd order theorem the attention was focussed on the prime 2. However, in 1976 the ”Classification” was psychologically over and so in 1980 the Classification theorem could be claimed. By the end of 1983 all of the relevant papers were published, except the 800 - pages manuscript on quasithin groups by Mason, which Aschbacher later would find to be incomplete.