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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 . Then G is either (a) a of prime ; (b) an of degree n 5; (c) a finite simple ;≥ 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 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 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 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 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 . | |

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 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.

4 4 The Classification theorem

In 1980 one had reached so far that the classification theorem could be claimed. As mentioned in the introduction, the extensive work later showed to be incom- plete, but maybe Aschbacher and Smith have the last (?) missing piece? We state the theorem again here.

Theorem 8 (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;≥ (d) one of 26 sporadic finite simple groups.

4.1 Cyclic groups of prime order A cyclic group G of prime order has only two subgroups at all, the trivial subgroup e and G itself (every element except the unit generates G). Hence, all cyclic groups{ } of prime order are simple. Cyclic groups of non-prime order are not simple. Assume that G = mn, m, n > 1,andthata generates G.Then m 2m (n 1)m | | a ,a , ..., a − is a non-trivial proper of G. If G is an abelian simple group, then G is cyclic of prime order. Assume ©that G = e is an abelianª group, then the subgroup generated by an element a = e is6 a{ non-trivial} normal subgroup of G.IfG is simple, this means that a must6 generate G and furthermore, a (and so G)mustbeofprimeorder.

4.2 Alternating groups of degree n 5 ≥ The alternating group An is the subgroup of the of n letters Sn (the group of all permutations of 1, 2, ..., n ) consisting of all even permutations. { } The alternating group An is simple for n 5. ≥ 4.3 Finite simple groups of Lie-Type A Lie-Type group is a finite analog of a Lie group. The Lie-Type groups can be split in two groups: the classical groups and the exceptional groups.

4.3.1 Classical groups The Chevalley and twisted Chevalley groups include the four classical families of linear simple groups (note that some groups can - at least up to isomorphisms -belongtomorethanonefamily).

The projective PSLn(q). •

One obtains the group PSLn(q) by factoring SLn(q) by the scalar matrices (matrices of the form λI) contained in SLn(q).

5 SLn(q) is the group of all invertible n n matrices, having elements in a finite field containing q elements, with determinant× =1. The group PSLn(q) is simple except for

PSL2(2) ∼= S3 and PSL2(3) ∼= A4.

The projective PSpn(q). •

One obtains the group PSpn(q) by factoring Spn(q) by the scalar matrices contained in Spn(q). AmatrixA is symplectic if

0 I AJAT = J,whereJ = n . In 0 µ − ¶ Spn(q) is the group of all 2n 2n symplectic matrices, having elements in a finite field containing q elements.× The group PSpn(q) is simple except for

PSp2(2) ∼= S3, PSp2(3) ∼= A4 and PSp4(2) ∼= S6.

The projective special PSUn(q). • One obtains the group PSUn(q) by factoring SUn(q) by the scalar matrices contained in SUn(q).

SUn(q) is the group of all n n unitary matrices (the invertible n n 1 × × matrices U fulfilling U − = U ∗,whereU ∗ is the hermitian conjugate of U), having elements in a finite field containing q elements, with determinant =1. The group PSUn(q) is simple except for

PSU2(2) ∼= S3, PSU2(3) ∼= A4 and PSU3(2).

The projective special PSOn(q). • One obtains the group PSOn(q) by factoring SOn(q) by the scalar matri- ces contained in SOn(q).

SOn(q) is the group of all n n orthogonal matrices (the invertible n n 1 T × × matrices A fulfilling A− = A ), having elements in a finite field containing q elements, with determinant =1. In general PSOn(q) is not simple. Actually, the structure of the group depends on the choice of quadratic form. Over a finite field (in contrast to the complex field C) there are two inequivalent non-degenerate symmetric bilinear forms. The corresponding orthogonal groups are distinct if n is even and so for even n we have two distinct groups, denoted + SOn (q) and SOn−(q), respectively. However, for odd n these groups are the same and so the notation SOn(q) is without contradictions. The derived groups (the derived group or the commutator subgroup of a group G is the subgroup of + G generated by the commutators of the elements in G)ofSO2n+1(q),SO2n(q)

6 + and SO2−n(q) are denoted Ω2n+1(q), Ω2n(q) and Ω2−n(q), respectively. We obtain + + the groups P Ω2n+1(q),PΩ2n(q) and P Ω2−n(q) by factoring Ω2n+1(q), Ω2n(q) + and Ω2−n(q) by the scalar matrices contained in Ω2n+1(q), Ω2n(q) and Ω2−n(q), respectively. The group P Ω2n+1(q) is simple except for

P Ω3(2) ∼= S3, PSL2(3) ∼= A4 and P Ω5(2) ∼= S6. + + Furthermore, the groups P Ω2n(q) and P Ω2−n(q) are simple for all n 3. P Ω4 (q) 2 ≥ ' PSL2(q) PSL2(q) is not simple and P Ω− (q) PSL2(q ) is simple. × 2n ' 4.3.2 Exceptional groups The exceptional groups can be split in two types of groups: the untwisted excep- tional Chevalley groups and the twisted exceptional Chevalley groups (including the Tits group). The simple untwisted exceptional Chevalley groups are

En(q), n 6, 7, 8 ; • ∈ { } F4(q); • G2(q), q 3; • ≥ The twisted Chevalley groups are certain subgroups of the Chevalley groups whose elements are fixed by some automorphisms (coming from symmetries of the Dynkin diagrams). The simple twisted exceptional Chevalley groups (including the Tits group) are (the superscript on the left denotes the order of the automorphism of the Dynkin diagram symmetry used in the construction)

3 D4(q); • 2 E6(q); • 2 2n+1 B2(2 ), n 1, Suzuki groups; • ≥ 2 2 F4(2)0, a subgroup of F4(2) called Tits group; • 2 2n+1 F4(2 ), n 1, Ree groups; • ≥ 2 2n+1 G2(3 ), n 1, Ree groups. • ≥ 4.4 Sporadic groups The sporadic groups are 26 (at least if the proof of about 5000 pages is not in- correct) and were discovered in-between 1966 and 1976, except the five Mathieu groups, that were discovered in the 1860’s. The sporadic groups are

Mathieu groups M11,M12,M22,M23,M24; •

7 Janko groups J1,J2,J3,J4; • Conway groups Co1,Co2,Co3; • Fischer groups Fi22,Fi23,Fi0 ; • 24 Highman-Sims group HS; • McLaughlin group McL; • Held group He; • Rudvalis group Ru; • Suzuki Suz; • O’Nan group O0N; • Harada-Norton group HN; • Lyons group Ly; • Thompson group Th; • Baby B; • Monster group M. • 4.4.1 The Monster and the Baby Monster The two largest sporadic groups are the Baby Monster group, whose order is approximately 4 1033, and the Monster group, whose order is 246 320 59 76 112 133 17 · 19 23 29 31 41 47 59 71 or approximately 8· 1053·.In· the· 1970’s· Sims· developed· · · powerful· · methods· · · for the computer construction· of finite simple groups as permutation groups and Sims managed to construct the Baby Monster, but the Monster was too big even for him. However, in 1980 Griess managed to construct the Monster group in a computer-free way. Griess defined a non-associative commutative algebra of dimension 196884, having the Monster as its group of automorphisms. The studies of the Griess algebra and its associated vertex operator algebra have stimulated some fruitful research. One of the mathematicians taking part in this research, Richard Borcherds, invented the notion of vertex algebras and also proved the so-called ”” conjecture (conjectured by John Conway and Simon Norton in the end of the 1970’s) in 1989, which provides an interrelationship between the Monster group and modular functions. The name of the conjecture comes from the totally unexpected relationship between these two mathematical structures. Borcherds was awarded the Fields Medal in 1998, much thanks to his work connected with the Monstrous Moonshine conjecture.

8 References

[1]R.W.Carter,Simple Groups of Lie Type, John Wiley & Sons, 1972. [2] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R.A. Wilson, Atlas of Finite Groups, Clarendon Press, 1985. [3]J.B.Fraleigh,A First Course in Abstract Algebra, Addison-Wesley, 4th ed., 1989. [4] D. Gorenstein, Finite Simple Groups, Plenum Press, 1985. [5] G. Gramble, Finite simple groups, http://www.itee.uq.edu.au/~gregg/finsimgps.ps.gz, 1996. [6] C. S. Rajan, Richard E. Borcherds, http://log24.com/log/saved/030528- Borcherds.html . [7] R. Solomon, A brief history of the classification of the finite simple groups, Amer. Math. Soc., 38 (2001), pp.315-352. [8] I. Stewart, Galois Theory, Chapman & Hall, 2nd ed., 1989. [9] http://www.berkeley.edu/news/media/releases/98legacy/08-19- 1998a.html [10] http://mathworld.wolfram.com/ClassificationTheoremofFiniteGroups.html [11] http://www.encyclopedia4u.com/c/classification-of-finite-simple- groups.html [12] http://www.nationmaster.com/encyclopedia/Classification-of-finite- simple-groups

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