sign in sign up
<<
Home , 3D4, Almost simple group, Alternating group, ATLAS of Finite Groups, Conway group, Conway group Co2

Introduction to finite simple groups

808017424794512875886459904961710757005754368000000000

246 320 59 76 112 133 17 19 23 29 31 41 47 59 71 · · · · · · · · · · · · · · 0.1

Contents

1. Introduction •

2. Alternating groups •

3. Linear groups • 4. Classical groups • 5. Chevalley groups • 6. Exceptional groups •

7. Sporadic groups: old generation • 8. Sporadic groups: new generation • 0.2

Literature

R. Wilson: The finite simple groups, ◦ Graduate Texts in 251, Springer, 2009. J. Conway, R. Curtis, R. Parker, ◦ S. Norton, R. Wilson: Atlas of finite groups, Clarendon Press Oxford, 1985/2004. P. Cameron: groups, ◦ LMS Student Texts 45, Cambridge, 1999. D. Taylor: The geometry of , ◦ Heldermann, 1992. R. Carter: Simple groups of Lie type, ◦ Wiley, 1972/1989. M. Geck: An introduction to algebraic geometry ◦ and algebraic groups, Oxford, 2003. R. Griess: Twelve sporadic groups, ◦ Springer Monographs in Mathematics, 1989.

Aim: Explain the statement of the CFSG: • 1.1

Classification of finite simple groups (CFSG)

Cyclic groups of prime C ; p a prime. • p Alternating groups ; n 5. • An ≥ Finite groups of Lie type: • Classical groups; q a prime power: ◦ Linear groups PSL (q); n 2, (n, q) = (2, 2), (2, 3). n ≥ 6 Unitary groups PSU (q2); n 3, (n, q) = (3, 2). n ≥ 6 Symplectic groups PSp (q); n 2, (n, q) = (2, 2). 2n ≥ 6 Odd-dimensional orthogonal groups Ω (q); n 3, q odd. 2n+1 ≥ + Even-dimensional orthogonal groups PΩ (q), PΩ− (q); n 4. 2n 2n ≥ Exceptional groups; q a prime power, f 1: ◦ ≥ E (q). E (q). E (q). F (q). G (q); q = 2. 6 7 8 4 2 6 2 2 3 3 Steinberg groups (q ). Steinberg groups D4(q ). 2 2f+1 2 2f+1 B2(2 ). Small Ree groups (3 ). 2 2f+1 2 Large Ree groups (2 ), Tits F4(2)′.

26 Sporadic groups: ... • 1.2

Classification of finite simple groups (CFSG), II

Sporadic groups: • Mathieu groups M , M , M , M , M . ◦ 11 12 22 23 24 Leech groups: ◦ Conway groups Co1, Co2, Co3. McLaughlin group McL. Higman-Sims group HS.

Suzuki group Suz. Hall- J2.

Fischer groups Fi , Fi , Fi′ . ◦ 22 23 24 Monstrous groups: ◦ Fischer-Griess Monster M. Baby Monster B. Thompson group Th. Harada-Norton group HN. He. Pariahs: ◦ Janko groups J1, J3, J4. O’Nan group ON. Ly. Ru.

Repetitions: • PSL (4) = PSL (5) = ; PSL (7) = PSL (2); ◦ 2 ∼ 2 ∼ A5 2 ∼ 3 PSL (9) = ; PSL (2) = ; ◦ 2 ∼ A6 4 ∼ A8 PSU (2) = PSp (3). ◦ 4 ∼ 4 1.3

Composition series

Let G be a finite group. ◦ G is called simple if G is non-trivial and does not have any proper • non-trivial normal .

Composition series: • G has a composition series of length n N ◦ ∈ 0 1 = G ⊳ G ⊳ ⊳ G = G, { } 0 1 ··· n

where Gi 1⊳Gi such that Gi/Gi 1 is simple, for all i 1,...,n . ◦ − − ∈{ } Jordan-H¨older : • The of composition factors Gi/Gi 1, counting multiplici- ◦ − ties, is independent of the choice of a composition series.

G is called soluble if all composition factors Gi/Gi 1 are abelian, • − or equivalently cyclic of prime order.

Examples: • 1 ⊳ with composition factors C . ◦{ } S2 2 1 ⊳ ⊳ with composition factors C ,C . ◦{ } A3 S3 2 3 1 ⊳ C ⊳ V ⊳ ⊳ with composition factors C ,C ,C ,C . ◦{ } 2 4 A4 S4 2 2 2 3 1 ⊳ ⊳ with composition factors ,C . ◦{ } A5 S5 A5 2 1.4

Some history

Abel’s Theorem: • The of the general equation of degree ◦ n N over any field is isomorphic to the symmmetric group . ∈ Sn The general polynomial equation of degree n N over a field of ◦ ∈ characteristic 0 is solvable by radicals if and only if its Galois group is soluble, that is if and only if n 4. ≤

Galois [ 1830]: simple for n 5, PSL (p) for p a prime. ◦ ∼ An ≥ 2 Jordan [1870]: ‘Trait´edes substitutions’, PSL (p). ◦ n Sylow [1872]: the first classification tool. ◦ Mathieu [1861/1873]: the simple Mathieu groups. ◦ Killing [ 1890]: classification of complex simple Lie algebras. ◦ ∼ Dickson [ 1900]: finite field analoga of the classical Lie groups. ◦ ∼ Chevalley [1955]: uniform construction of ◦ the classical and exceptional finite groups of Lie type. Ree, Steinberg, Suzuki, Tits [ 1960]: ◦ ∼ twisted classical and exceptional finite groups of Lie type. 1960: common belief is that all finite simple groups are known. ◦∼ 1.5

Some history, II

Brauer, Fowler [1955]: ◦ Given n N, there are at most finitely many simple groups contain- ∈ ing an with centraliser of order n. Feit-Thompson Theorem [1963]: ◦ Any finite group of odd order is soluble. Brauer program: Hence any non-abelian finite ◦ contains an involution, thus consider centralisers of central involu- tions and prove completeness of classification by induction. Janko [1964]: (the first since almost a century) ◦ J with involution centraliser C . 1 2 ×A5 Thompson [1968]: classification of minimal simple groups. ◦ Janko [1975]: the last sporadic group J . ◦ 4 1980: common belief is that CFSG is proved. ◦∼ Gorenstein, Lyons, Solomon [ 1994]: ◦ ≥ revision project of the proof of CFSG. Aschbacher, Smith [2004]: ◦ the quasithin case, completing the proof of CFSG.

Do we really believe that the Four-Colour Theorem, or Fer- • mat’s Last Theorem, or the Poincar´eConjecture, or the CFSG are proved? 1.6

Applications of CFSG

Let T be a non-abelian finite simple group. • Then Z(T ) = 1 implies T = Inn(T ) E Aut(T ). ◦ { } ∼ A group G such that T G Aut(T ) is called almost simple. • ≤ ≤ A G such that G/Z(G) = T is called quasi-simple. • ∼

Schreier’s Conjecture: • The outer group Out(T ) := Aut(T )/Inn(T ) of any ◦ finite simple group T is soluble. Proof: by inspection; in all cases Out(T ) is ‘very small’. ♯ •

Theorem: Let N E G such that gcd( N , G/N ) = 1. Then all • | | | | complements of N in G are conjugate.

Proof: uses the Feit-Thompson Theorem; or alternatively: • Let G = N : H be a minimal counterexample. ◦ Easy: N is non-abelian simple and C (N) = 1 ◦ G { } Hence G = G/C (N) Aut(N) such that N Inn(N). ◦ ∼ G ≤ ≤ Thus G/N Out(N) is soluble. ◦ ≤ Hence the assertion follows from Zassenhaus’s Theorem. ♯ ◦ 1.7

Applications of CFSG, II

Multiply-transitive permutation groups: • The finite 2-transitive groups are explicitly known. ◦ The only finite 6-transitive groups are symmetric and alternating. ◦ The only finite 4-transitive groups are symmetric and alternating, ◦ and the Mathieu groups M11, M12, M23, and M24.

Proof: • Burnside’s Theorem: A minimal non-trivial ◦ of a finite 2-transitive group is either elementary-abelian and regular, or simple and primitive. Hence a 2-transitive group is either affine or almost simple: ◦ Huppert and Hering: soluble and insoluble affine cases; ◦ Maillet, Curtis, Kantor, Seitz, Howlett: ◦ almost simple cases. The higher transitive groups are then found by inspection. ♯ ◦ Example: • ASL (q) = [qd]: SL (q), where q is a prime power and n = qd. ◦ d ∼ d qd 1 PSLd(q), where q is a prime power, d 2, and n = q −1 . ◦ ≥ − 2.1

Symmetric and alternating groups

Let n N . ◦ ∈ 0 Let be the on 1,...,n . • Sn { } Let sgn: 1 = C be the sign representation. ◦ Sn → {± } ∼ 2 Let := ker(sgn)E be the on 1,...,n ; ◦ An Sn { } the elements of are called even , ◦ An the elements of are called odd permutations. ◦ Sn \An The cycle type of a permutation is the partition of n indicating • the lengths of its distinct cycles, counting multiplicities. Example: The identity has cycle type [1n], ◦ n 2 a2-cycle or transposition has cycle type [2, 1 − ], n 3 a3-cycle has cycle type [3, 1 − ]. A permutation is even if and only if it has an even number of cycles ◦ of even length.

The conjugacy classes of are parametrised by cycle types. • Sn A permutation is centralised by no odd permutation if and only ◦ if it is the product of cycles of distinct odd lengths. Hence the orbit-stabiliser theorem implies: ◦ A of contained in splits into two conjugacy ◦ Sn An classes of if and only if its cycle type has pairwise distinct odd An parts, otherwise it is a single conjugacy class of . An 2.2

Simplicity of An

Theorem: Let n 5. Then is simple. • ≥ An Proof: by induction on n; let 1 = N E . • { } 6 An Let n = 5. Then N is a union of conjugacy classes. • The cycle types of even permutations are [15], [3, 12], [22, 1], [5], ◦ where only type [5] splits into two conjugacy classes. The conjugacy class lengths are 1, 20, 15, 12, 12, respectively. ◦ No sub-sum of these, strictly including 1, divides 60; thus N = . ◦ An

Let n > 5. Then n 1 = Stab n(n) is simple. • A − A N n 1 E n 1, hence i) n 1 N or ii) N n 1 = 1 : ◦ ∩A − A − A − ≤ ∩A − { } n 3 i) Then N contains all elements of cycle type [3, 1 − ]. Any even permutation is a product of 3-cycles; thus N = . ◦ An ii) Then any non-trivial of N acts fixed-point-free. σ τ 1 If 1 =1 for σ, τ N, then στ − N n 1 = 1 . ◦ ∈ ∈ ∩A − { } Thus N n. ◦ | |≤ But does not have a non-trivial conjugacy class with fewer than ◦ An n elements, a contradiction. ♯ 2.3

Automorphisms of An

Let n 4. Then Z( ) = 1 , hence = Inn( ) E Aut( ); ◦ ≥ An { } An ∼ An An and acts faithfully by conjugation, hence Aut( ). ◦ Sn Sn ≤ An Theorem: Let n 7. Then Aut( ) = . • ≥ An Sn Proof: [C. Parker] • being simple, it cannot possess a proper subgroup of index ◦ An k

We show ( ): If n 1 = H < n, then H = Stab n(i) for some i. • ∗ A − ∼ A A Let n = 7. H cannot have a non-trivial orbit of less than 6 points. ◦ If H is not a point stabiliser, then H acts transitively on 1,..., 7 . { } This is a contradiction since 7 H = , proving ( ) for n = 7. 6| | | |A6| ∗

Let n 9. A ‘3-cycle’ of H centralises a group = n 4. ◦ ≥ ∼ A − Since n 4 5 the latter has an orbit of at least n 4 points. − ≥ − Thus a ‘3-cycle’ of H moves at most 4 points, thus is a 3-cycle of . An Let n = 8. A ‘3-cycle’ of H centralises a group = . ◦ ∼ A4 Hence there is a V4 centralising the ‘3-cycle’. The elements of of cycle type [32, 12] do not centralise a V . A8 4 Hence a ‘3-cycle’ of H is a 3-cycle of . A8 2.4

Automorphisms of , II An

Thus for n 8 the ‘3-cycles’ of H map to 3-cycles of . ◦ ≥ An For pairs of 3-cycles we have (a,b,c), (a,b,d) = . ◦ h i ∼ A4 Hence the subgroup ◦ H = n 1 = (1, 2, 3),..., (1, 2,n 1) ∼ A − h − i maps to a subgroup

(a,b,c1),..., (a,b,cn 3) n. h − i≤A The latter moves n 1 points. ◦ − Hence H Stab n(i) for some i, proving ( ) for n 8. ◦ ≤ A ∗ ≥ Now: • Any automorphism permutes the isomorphic to n 1. ◦ A − These subgroups are in natural with 1,...,n . ◦ { } Hence any automorphism induces a permutation of 1,...,n . ♯ ◦ { }

We have Aut( ) = for n 4, 5 . • An Sn ∈{ } We have Aut( ) = .22. ◦ A6 ∼ A6 has two conjugacy classes of subgroups isomorphic to . ◦A6 A5 2.5

Schur covers of and Sn An

A finite group H such that Z(H) H′ and H/Z(H) = G is • ≤ ∼ called an Z(H) -fold cover of G. | | Two maximal covers of G are isoclinic. ◦ If G is perfect, its unique maximal cover is a universal cover. ◦

has maximal 2-fold covers =2. , for n 4, •An An An ≥ except for n 6, 7 where it hase maximal 6-fold covers 6. n. ◦ ∈{ } A has two maximal 2-fold covers and , for n 4, •Sn Sn Sn ≥ both of shape 2. n, but we have ne= n ifb and only if n = 6. ◦ S S ∼ S e b

The Coxeter presentation of , where n N, is given as • Sn ∈ 2 3 2 n = s1,...,sn 1 si =(sisi+1) =(sisj) = 1 for i j 2 , S ∼ h − | | − |≥ i where adjacent transpositions (i,i + 1) s . ◦ 7→ i For and , where n 4, we have [Schur, 1911]: • Sn Sn ≥ e n :=b s1,...,sn 1, z S h − | z2 =1, s2 =(s s )3 = z, (s s )2 = z e i i i+1 i j i n := s1,...,sn 1, z S h − | z2 =1, s2 =(s z)2 =(s s )3 = 1, (s s )2 = z b i i i i+1 i j i 2.6

Subgroups of Sn

Describing all the subgroups of , for all n N , is by ◦ Sn ∈ 0 Cayley’s Theorem equivalent to classifying all finite groups: ◦ hopeless. ◦ But there are certainly are interesting prominent subgroups: ◦ for example, intransitive subgroups. • Partition the set of n = km points into m blocks of size k. • The = m : acts on this partition, ◦ Sk ≀Sm ∼ Sk Sm where the base group m = consists of permutations ◦ Sk Sk×···×Sk of the various blocks, and the wreathing permutes the blocks. ◦ Sm < is an imprimitive transitive subgroup, for k,m 2. ◦Sk ≀Sm Sn ≥ acts on 1,...,k m by the product action, n = km, •Sk ≀Sm { } where [π ,...,π ] m acts by [a ,...,a ] [aπ1,...,aπm], ◦ 1 m ∈Sk 1 m 7→ 1 m 1 and π− acts by [a ,...,a ] [a π ,...,a π ]. ◦ ∈Sm 1 m 7→ 1 m < is a primitive subgroup, for k 3 and m 2. ◦Sk ≀Sm Sn ≥ ≥ 2.7

Maximal subgroups of Sn

One might try to describe the maximal subgroups of ; ◦ Sn the maximal subgroups of are then obtained by intersection: ◦ An O’Nan-Scott Theorem [1979]: Any proper subgroup of • Sn different from is contained in one of the following subgroups: An i) an intransitive group , where n = k + m; Sk ×Sm ii) an imprimitive transitive group , where n = km; Sk ≀Sm iii) a primitive wreath product , where n = km; Sk ≀Sm d d iv) an affine group AGLd(p) ∼= p : GLd(p), where n = p ; v) a diagonal type group T m.(Out(T ) ) = (T ).Out(T ), ×Sm ∼ ≀Sm where T is a non-abelian simple group, m 1 acting on the of a subgroup of index n = T − , of shape | | ∆(T ).(Out(T ) ) = Aut(T ) ; ×Sm ∼ ×Sm vi) an , acting on the cosets of a of index n.

Describing the groups in class vi) requires complete knowledge of ◦ the maximal subgroups of all almost simple groups: reducing an impossible problem to an even harder one. ◦ 3.1

Linear groups

Let F be the field with q = pf elements, p a prime, f N, n N. ◦ q ∈ ∈ n n General GL (q) := g F × ; det(g) =0 • n { ∈ q 6 } Counting the number of ordered F -bases of Fn: ◦ q q n n n n n 1 (2) n i GL (q) =(q 1)(q q) (q q − ) = q (q 1) ◦| n | − − ··· − · i=1 − Viewing q as an indeterminate, Q ◦ this is an order polyomial in Z[q], ◦ whose irreducible factors are q and cyclotomic . ◦ SL (q) := g GL (q); det(g)=1 • n { ∈ n } Projective PGL (q) := GL (q)/Z(GL (q)), • n n n where Z(GLn(q)) = Fq∗ En = Cq 1. ◦ · ∼ − 1 SLn(q) = PGLn(q) = q 1 GLn(q) ◦| | | | − ·| | Projective special linear group PSL (q) := SL (q)/Z(SL (q)), • n n n n where Z(SLn(q)) = λ En; λ =1 = Cgcd(n,q 1). ◦ { · } ∼ − 1 1 1 PSLn(q) = gcd(n,q 1) SLn(q) = gcd(n,q 1) q 1 GLn(q) ◦| | − ·| | − · − ·| | 3.2

Simplicity of PSLn(q)

PSL (2) = GL (2) = : • 2 ∼ 2 ∼ S3 GL (2) acts 2-transitively on the three vectors in F2 0 . ◦ 2 2 \{ } PSL (3) = : • 2 ∼ A4 GL (3) acts on the four 1-dimensional F -subspaces of F2, ◦ 2 3 3 1 0 the action is 2-transitive, fixes the standard F -basis, ◦ 0 1 3 − hence GL (3) , with Z(GL (3)) = C , ◦ 2 →S4 2 ∼ 2 thus PGL (3) = and PSL (3) = . ◦ 2 ∼ S4 2 ∼ A4 Note: GL (3) = and SL (3) = . ◦ 2 ∼ S4 2 ∼ A4 e e

Theorem: Let n 2 and (n, q) = (2, 2), (2, 3). • ≥ 6 Then PSLn(q) is simple.

Proof: • G := SL (q) acts on the set of 1-dimensional subspaces of Fn, ◦ n q yielding a 2-transitive, hence primitive, action of PSL (q). ◦ n Let x := [1, 0,..., 0] F and H := Stab (x), ◦ h i q G then ◦ λ 0n 1 H = − G; λ Fq∗,h GLn 1(q),λ det(h)=1 .  h  ∈ ∈ ∈ − ·  ∗ 3.3

Simplicity of PSLn(q), II

Use Iwasawa’s Criterion: ◦ Let ◦ 1 0n 1 A := − H ,  En 1 ∈  ∗ − then A ⊳ H is abelian, consisting of transvections, ◦ that is g G such that rk(g E ) = 1 and rk((g E )2)=0. ◦ ∈ − n − n Jordan normal form theorem implies that ◦ any transvection is G-conjugate to some element of A. • G is generated by transvections: • Any g G can be reduced to E by a sequence of elementary row ◦ ∈ n operations of the form ‘r r + λr ’, i 7→ i j that is multiplying g from the right with a series of transvections. ◦ G is perfect: • For n 3 any transvection is a : ◦ ≥ 1 0 0 1 0 0 1 00 1 1 0 , 0 1 0 =  0 10 0 0 1 0 λ 1 λ 0 1     −  2 For n = 2 and q 4 there is λ F∗ such that λ = 1, then ◦ ≥ ∈ q 6 1 0 λ 0 1 0 , 1 = 2 β 1 0 λ−  β(λ 1) 1 − is an arbitrary element of A. ♯ 3.4

Iwasawa’s Criterion

Theorem: [Iwasawa, 1941] • Let G be a finite group, acting primitively on a set Ω, ◦ let H := Stab (x)

Despite its simplicity this is astonishingly powerful. ◦ Exercise: Use it to prove the simplicity of , for n 5. ◦ An ≥ 3.5

Automorphisms of SLn(q)

Diagonal automorphisms: • induced by conjugation with diagonal matrices, ◦ that is by the conjugation action of GL (q). ◦ n GLn(q)/SLn(q) = Cq 1, PGLn(q)/PSLn(q) = Cgcd(n,q 1) ◦ ∼ − ∼ − automorphisms: • induced by the Frobenius automorphism ϕ : λ λp of F , ◦ p 7→ q where q = pf , hence ϕ = C . ◦ h pi ∼ f Semilinear groups ◦ ΓL (q) := GL (q): ϕ , PΓL (q) := PGL (q): ϕ , n n h pi n n h pi ΣL (q) := SL (q): ϕ , PΣL (q) := PSL (q): ϕ . n n h pi n n h pi Graph automorphisms: • induced by a graph automorphism of the . ◦ tr Duality GL (q) GL (q): g g− ; ◦ n → n 7→ induces duality on SL (q), PGL (q), PSL (q). ◦ n n n Note: duality is not inner for n 3. ◦ ≥

These are all the ‘outer’ automorphisms; • in particular the outer is soluble. ◦ 3.6

Covers of PSLn(q)

PSL (q) has gcd(n, q 1)-fold universal cover • n − SLn(q) = Cgcd(n,q 1).PSLn(q), ∼ −

except: • PSL (4) = PSL (5) = has universal cover 2.PSL (4); ◦ 2 ∼ 2 ∼ A5 2 PSL (9) = has universal cover 6.PSL (9); ◦ 2 ∼ A6 2 PSL (2) = PSL (7) has universal cover 2.PSL (2); ◦ 3 ∼ 2 3 PSL (2) = has universal cover 2.PSL (2); ◦ 4 ∼ A8 4 PSL (4) has universal cover (3 42).PSL (4). ◦ 3 × 3

Note: • generic universal covers have order coprime to the ◦ defining characteristic p of the Lie type group, while exceptional parts of universal covers are p-groups. ◦ 3.7

Subgroups of GLn(q)

Borel subgroup B := g; g lower triangular

Axiomatic: BN-pairs [Tits, 1962] • 3.8

Maximal subgroups GLn(q)

Aschbacher-Dynkin Theorem: [1984/1952] • Any proper subgroup of GL (q) different from SL (q) is contained ◦ n n in one of the following subgroups: i) a reducible group qkm : (GL (q) GL (q)), where n = k + m, k × m the stabiliser of a k-dimensional Fq-subspace; ii) an imprimitive group GL (q) , where n = km, k ≀Sm the stabiliser of a direct sum decomposition into m k-subspaces; iii) a tensor product GL (q) GL (q), where n = km, k ◦ m the stabiliser of a tensor product decomposition Fk Fm; q ⊗ q iv) a wreathed tensor product, the preimage in GL (q) of PGL (q) , where n = km, n k ≀Sm the stabiliser of a tensor product decomposition Fk Fk; q ⊗···⊗ q 2k k v) the preimage in GLn(q) of r : Sp2k(r), where n = r , or of 22k.GOǫ (2), for r = 2 and q ǫ (mod 4); 2k ≡ vi) an almost quasi-simple group acting irreducibly.

Aschbacher: looks more closely at case vi) , • in particular considers subfields and extension fields of F . ◦ q 3.9

Proof of the Aschbacher-Dynkin Theorem

Proof: • Let PSL (q) H

Geometric algebra

Let F be a field, with automorphism σ : F F such that σ2 = id, ◦ → and let V be a finitely generated F -. ◦ A σ-bilinear form is a map f : V V F such that • × → f(λu + v, w) = λf(u, w) + f(v, w), ◦ f(u,λv + w) = λσf(u,v) + f(u, w). ◦ f is called • symmetric if σ = id and f(w,v) = f(v, w), ◦ hermitian if σ = id and f(w,v) = f(v, w)σ, ◦ 6 symplectic if σ = id and f(w,v) = f(v, w), ◦ − alternating if σ = id and f(v,v)=0. ◦ Any alternating form is symplectic, ◦ if char(F ) = 2 then any symplectic form is alternating; ◦ 6 if char(F ) = 2 then being symmetric or symplectic coincide. ◦ A quadratic form is a map q : V F such that • → q(λv + w) = λ2q(v) + q(w) + λf(v, w), ◦ where the associated bilinear form f : V V F is symmetric. ◦ × → If char(F ) = 2 then q is recovered from f as q(v) = 1f(v,v), ◦ 6 2 if char(F ) = 2 then f is alternating. ◦ 4.2

Geometric algebra, II

A σ-bilinear form f is called non-degenerate, if • rad(f) := w V ; f(v, w) =0 for all v V = 0 . { ∈ ∈ } { } v V is called isotropic if f(v,v)=0. ◦ ∈ A map A GL(V ) is called an isometry of f, if ◦ ∈ f(vA, wA) = f(v, w) for all v, w V ; ∈ the set of all isometries is a subgroup of GL(V ). ◦

A quadratic form q is called non-degenerate, if • rad(q) := v rad(f); v singular = 0 , { ∈ } { } where v V is called singular if q(v)=0. ◦ ∈ The Witt index is the dimension of a maximal singular subspace; ◦ by Witt’s Theorem this is independent of the subspace chosen. ◦ A map A GL(V ) is called an isometry of q, if ◦ ∈ q(vA) = q(v) for all v V ; ∈ the set of all isometries is a subgroup of GL(V ). ◦

No classification of non-degenerate forms for arbitrary F is known. • 4.3

Unitary groups

Theorem: Any non-degenerate ϕ -hermitian form over F 2 has • q q an orthonormal Fq2-basis, that is the associated Gram is E . ◦ n Thus g GL (q2) is an isometry if and only if g E gtr = E . • ∈ n · n · n 2 2 tr General GU (q ) := g GL (q ); g− = g , ◦ n { ∈ n } that is the fixed points of the concatenation of the graph auto- ◦ 2 morphism (the duality) and a field automorphism of GLn(q ).

Counting the number of ordered orthonormal F 2-bases: • q n n GU (q2) = q(2) n (qi ( 1)i)=( q)(2) n (( q)i 1) ◦| n | · i=1 − − − · i=1 − − Ennola duality QGU (q2) = GL ( q) Q ◦ | n | | n − | As in the linear case: SU (q2), PGU (q2), PSU (q2), • n n n 2 where Z(GUn(q )) = Cq+1 = C ( q) 1 . ◦ ∼ | − − | PSU (q2) = 1 1 GU (q2) = PSL ( q) ◦| n | gcd(n,q+1) · q+1 ·| n | | n − | Simplicity of PSU (q2): Apply Iwasawa’s Criterion • n to the action on the set of isotropic 1-dimensional subspaces, ◦ and use unitary transvections, ◦ that is V V : v v + λf(v, w)w, where w V is isotropic. ◦ → 7→ ∈ Exceptions: PSU (q2) = PSL (q), and PSU (22) is soluble. ◦ 2 ∼ 2 3 4.4

Symplectic groups

Theorem: Any (necessarily even-dimensional) non-degenerate al- • ternating form over Fq is an orthogonal sum of hyperbolic planes; 0 1 that is the latter have Gram matrix . ◦  1 0 − Sp (q) • 2n Counting the number of ordered symplectic Fq-bases: ◦ 2 Sp (q) = qn n (q2i 1) ◦| 2n | · i=1 − We have Sp (q)Q SL (q). ◦ 2n ≤ 2n Projective symplectic group PSp (q) := Sp (q)/Z(Sp (q)), • 2n 2n 2n where Z(Sp (q)) = E . ◦ 2n {± n} 1 PSp2n(q) = gcd(2,q 1) Sp2n(q) ◦| | − ·| | Simplicity of PSp (q): Apply Iwasawa’s Criterion • 2n to the action on the set of 1-dimensional subspaces, ◦ and use symplectic transvections, ◦ that is V V : v v + λf(v, w)w. ◦ → 7→ Exceptions: Sp (q) = SL (q), and Sp (2) = . ◦ 2 ∼ 2 4 ∼ S6 4.5

Orthogonal groups

Theorem: Any (2n + 1)-dimensional non-degenerate quadratic • 2 n form over Fq is equivalent to X0 + i=1 XiX i. − P Theorem: Any 2n-dimensional non-degenerate quadratic form • over Fq is equivalent n either to i=1 XiX i, having maximal Witt index n, ◦ − or to, whereP T 2 + T + a F [T ] is irreducible, ◦ ∈ q n 1 2 2 − (X0 + X0X 0 + aX 0) + XiX i, − − − Xi=1 having non-maximal Witt index n 1. − + General orthogonal groups GO (q), GO (q), GO− (q) • 2n+1 2n 2n Counting the number of isotropic vectors, • which are acted on transitively by GOn(q), and induction: ◦ n ǫ (2) n n 1 2i GO2n(q) =2q (q ǫ) i=1− (q 1) ◦| | 2· − · − GO (q) =2qn n (q2iQ 1), ◦| 2n+1 | · i=1 − Q As in the linear case: SO (q), PGO (q), PSO (q), • n n n where Z(GO (q)) = E , ◦ n {± n} and where g J gtr = J, for J being the Gram matrix, ◦ · · implies det(g)2 = 1 for all g GO (q). ◦ ∈ n But: PSO (q) is in general not perfect. • n 4.6

Orthogonal groups in odd characteristic

Let q be odd. ◦ 2 Spinor norm ν : GO (q) F∗/F∗ = C : • n → q q ∼ 2 write g GO (q) as a product of reflections ◦ ∈ n r : V V : v v f(v,w) w, where w V is non-singular, ◦ w → 7→ − q(w) · ∈ 2 2 and let ν(r ) := q(w) F∗ F∗/F∗ . ◦ w · q ∈ q q Note the similarity to the definition of the sign of a permutation. ◦ Let Ω (q) := ker(ν) SO (q) and PΩ (q):=Ω (q)/Z(Ω (q)), • n ∩ n n n n then GO (q)/ ker(ν) = SO (q)/Ω (q) = C . ◦ n ∼ n n ∼ 2 SO (q) = PSO (q) and Ω (q) = PΩ (q), • 2n+1 ∼ 2n+1 2n+1 ∼ 2n+1 hence Ω (q) = 1 GO (q) . ◦ | 2n+1 | 4 ·| 2n+1 | E Ωǫ (q) if and only if qn ǫ (mod 4), •− 2n ∈ 2n ≡ ǫ 1 ǫ hence PΩ2n(q) = 2 gcd(4,qn ǫ) GO2n(q) . ◦ | | · − ·| | Simplicity of PΩ (q): Apply Iwasawa’s Criterion • n to the action on the set of 1-dimensional singular subspaces, ◦ and use Siegel transformations. ◦ ǫ Exceptions: GO2(q) = D2(q ǫ), and PΩ3(3) = PSL2(3) = 4, and ◦ ∼ − ∼ ∼ A PΩ+(q) = PSL (q) PSL (q). 4 ∼ 2 × 2 Note: Ω (q) = PSp (q) , but Ω (q) = PSp (q). ◦ | 2n+1 | | 2n | 2n+1 6∼ 2n 4.7

Orthogonal groups in characteristic 2

Let q =2f . ◦ GO (q) = SO (q) = PGO (q) = PSO (q) ◦ n n n n Theorem: GO (q) = Sp (q) • 2n+1 ∼ 2n Hence only consider the even-dimensional case: ◦ Quasideterminant ν : GOǫ (q) 1 = C : • 2n → {± } ∼ 2 write g GOǫ (q) as a product of orthogonal transvections ◦ ∈ 2n t : V V : v v + f(v, w) w, where w V , ◦ w → 7→ · ∈ and let ν(t ) := 1. ◦ w − Kantor: Then ν(g) is the sign of the permutation induced by g ◦ on the set of maximal isotropic subspaces.

Let Ωǫ (q) := ker(ν). • 2n Then the order formulae and the simplicity proof are still valid; ◦ ǫ + the latter with the exceptions GO2(q) = D2(q ǫ), and PΩ4 (q) = ◦ ∼ − ∼ PSL (q) PSL (q), and PΩ (2) = Sp (2) = . 2 × 2 5 ∼ 4 ∼ S6

Note: For arbitrary q we have, using Klein correspondence, • ǫ GO2(q) = D2(q ǫ), PΩ3(q) = PSL2(q), ◦ ∼ − ∼ + 2 PΩ (q) = PSL (q) PSL (q), PΩ−(q) = PSL (q ), ◦ 4 ∼ 2 × 2 4 ∼ 2 + PΩ (q) = PSp (q), PΩ (q) = PSL (q), PΩ−(q) = PSU (q). ◦ 5 ∼ 4 6 ∼ 4 6 ∼ 4 4.8

Structure of classical groups

Subgroups: • groups with BN-pairs, ◦ tori, Borels, and parabolics described in terms of geometry; ◦ entailing a generic ‘Iwasawa type’ simplicity argument. ◦ Moreover: ◦ Automorphisms: • diagonal, field, and graph automorphisms ◦ Covers: • generic p′-fold covers, and finitely many p-power-fold exceptions ◦ Maximal subgroups: • Dynkin [1952]: complex classical groups ◦ Aschbacher [1984]: finite classical groups ◦ Kleidman, Liebeck [1990]: explicit lists ◦ 5.1

Modern view of classical groups

Linear and classical groups: described in terms of • geometry, ◦ Lie theory, ◦ algebraic groups. ◦ Example: SL (q) is described by • n its natural faithful action on the n-dimensional space Fn; ◦ q the conjugation action on the (n2 1)-dimensional Lie algebra ◦ − n n sl (q) := A F × ; Tr(A)=0 , n { ∈ q } yielding an action of PSLn(q)=SLn(q)/Z(SLn(q)); polynomial equations defining the ◦ n n SL (F) := A F × ; det(A)=1 , n { ∈ } where F F is an algebraic closure with Frobenius morphism q ⊆ F := ϕ : F F: λ λq, q → 7→ yielding the set of fixed points SL (q)=SL (F)F := g SL (F); F (g) = g . n n { ∈ n }

Starting point: Classification of simple complex Lie algebras • by Dynkin types A , B , C , D , E , E , E , F , G . ◦ n n n n 6 7 8 4 2 5.2

Chevalley groups

Chevalley [1955]: • integral forms of simple complex Lie algebras ◦ yield simple Lie algebras L over any field F ; ◦ consider adjoint representation ◦ ad: L End (L): x (L L: y [x, y]), → F 7→ → 7→ and integrate suitable roots x L, ◦ ∈ obtain one-parameter subgroups of Aut(L), given by ◦ λi exp(λ ad(x)) := ad(x)i GL (L). · i! · ∈ F Xi 0 ≥

Chevalley group • G (F ) := exp(λ ad(x)); x L root,λ F Aut(L) n h · ∈ ∈ i≤

This uniformly yields finite field analoga of • the classical Lie groups, ◦ and the exceptional groups G , F , E , E , E . ◦ 2 4 6 7 8 G (F ) is a group with BN-pair. • n 5.3

Chevalley group of type A1

sl (F ) = f,h,e , with Chevalley basis • 2 h iF 0 0 1 0 0 1 f := , h := , e := . 1 0 0 1 0 0 − Adjoint action of e is nilpotent: ◦ 01 0 0 0 2 2 − 3 ad(e) = 0 0 2 , ad(e) = 00 0  , ad(e) =0 E3. − · 00 0 00 0     Integration λ ad(e) and λ ad(f) is well-defined: ◦ · · 2 2 1 λ λ λ 2 − exp(λ ad(e)) = E3 + λ ad(e) + ad(e) = 0 1 2λ · · 2 · − 00 1   2 1 00 λ 2 exp(λ ad(f)) = E3 + λ ad(f) + ad(f) =  2λ 1 0 · · 2 · λ2 λ 1 − −  SL (F ) = x(λ), y(λ); λ F , with transvections • 2 h ∈ i 1 λ 1 0 x(λ) := , y(λ) := . 0 1 λ 1 Adjoint action of SL (F ) on sl (F ) is conjugation: ◦ 2 2 x(λ): f f + λh λ2e, h h 2λe, e e; 7→ − 7→ − 7→ y(λ): f f, h h +2λe, e λ2f λh + e. 7→ 7→ 7→ − Thus we have SL (F ) A (F ), implying • 2 → 1 A (F ) := exp(λ ad(e)), exp(λ ad(f)); λ F = PSL (F ). 1 h · · ∈ i ∼ 2 5.4

Twisted groups

Generalise the construction of unitary groups from linear groups, ◦ as fixed point sets under suitable graph automorphisms: ◦ completes the list of classical groups; • yields twisted exceptional groups • 2E (q2) and 3D (q3) [Steinberg, 1959]; ◦ 6 4 yields ‘sporadic’ twisted exceptional groups • 2B (22f+1) [Suzuki, 1962], ◦ 2 2G (32f+1) [Ree, 1961], ◦ 2 2F (22f+1) [Ree, Tits, 1961/1964]. ◦ 4 These also are groups with BN-pair. •

Are there geometrical interpretations of these groups? • Mostly there are, elucidating more of the group structure; ◦ and leading to natural representations ◦ smaller than the adjoint representations. ◦ For E (q) the smallest representation has dimension 56, • 7 while the adjoint representation has dimension 133. ◦ For E (q) the adjoint representation is smallest, of dimension 248. • 8 5.5

Classical Dynkin types

Six series of classical groups: ◦ Classical Chevalley groups: • 1 2 3n−1 n Type A : PSL (q), for n 1 ◦ n n+1 ≥ 1 2 3 n−1 n Type B : Ω (q), for n 3 ◦ n 2n+1 ≥ 1 2 3 n−1 n Type C : PSp (q), for n 2 ◦ n 2n ≥ n−1 1 2 3 n−2

+ Type D : PΩ (q), for n 4 n ◦ n 2n ≥ Twisted classical groups: • 1 2 3n−1 n Type 2A : PSU (q), for n 2 ◦ n n+1 ≥ n−1 1 2 3 n−2

2 Type D : PΩ− (q), for n 4 n ◦ n 2n ≥ 5.6

Exceptional Dynkin types

Ten series of exceptional groups: ◦ Exceptional Chevalley groups: • 1 2 3 n−2 n−1

Type E , for n 6, 7, 8 n ◦ n ∈{ } 1 2 3 4 Type F ◦ 4 1 2 Type G ◦ 2 Twisted exceptional groups: • 1 2 3 4 5

Type 2E (q2) 6 ◦ 6 3 1 2

Type 3D (q3) 4 ◦ 4 1 2 Type 2B (22f+1) ◦ 2 1 2 Type 2G (32f+1) ◦ 2 1 2 3 4 Type 2F (22f+1) ◦ 4 6.1

Suzuki groups

Let q := 22f+1 for f N . ◦ ∈ 0 Consider the exceptional = Sp (2) = B (2): • S6 ∼ 4 2 Natural permutation representation of over F := F ◦ S6 q has - form f([x ,...,x ], [y ,...,y ]) := 6 x y . ◦ S6 1 6 1 6 i=1 i i P Then V := v ⊥/ v , where v := [1,..., 1], ◦ h iF h iF has -invariant non-degenerate alternating form, ◦ S6 hence we have Sp (q); now compare orders for q = 2. ◦ S6 ≤ 4 V has hyperbolic basis • e1 := [1, 1, 0, 0, 0, 0], f1 := [0, 1, 1, 0, 0, 0], e2 := [0, 0, 0, 1, 1, 0], f2 := [0, 0, 0, 0, 1, 1].

2 Exterior V ′ := Λ (V ) has ◦ non-degenerate symplectic form f ′ (Klein correspondence) ◦ given by f ′(a b, c d) = 1 if and only if dim( a,b,c,d )=4. ◦ ∧ ∧ h iF v′ ⊥/ v′ , where v′ := e f + e f , has hyperbolic basis ◦h iF h iF 1 ∧ 1 2 ∧ 2 e′ := e e , f ′ := f f , e′ := e f , f ′ := e f . 1 1 ∧ 2 1 1 ∧ 2 2 1 ∧ 2 2 2 ∧ 1

γ : e e′, f f ′ defines a graph automorphism of Sp (q) • i 7→ i i 7→ i 4 such that γ2 = ϕ , hence (γϕf )2 = ϕ1+2f = id. ◦ 2 2 2 Suzuki group Sz(q) := 2B (q) := C (γϕf ) [Ono, 1962] • 2 Sp4(q) 2 Note: γ extends < = Sp (2) to PGL (9) = . ◦ A6 S6 ∼ 4 2 6∼ S6 6.2

Suzuki groups, II

Sz(q) acts 2-transitively on the Tits oval [Suzuki, 1962], • a certain set of q2 + 1 many 1-dimensional subspaces of V , ◦ 1+1 with point stabiliser q : Cq 1, ◦ − whose central involutions are and generate Sz(q). ◦ This yields Sz(q) =(q2 + 1)q2(q 1), ◦ | | − and Iwasawa’s Criterion implies simplicity, ◦ with the exception Sz(2) = 5: 4. ◦ ∼ Automorphisms: only field automorphisms • Covers: generically trivial, • with the exception 22.Sz(8). ◦ Maximal subgroups, for f 1: [Suzuki] • ≥ 1+1 q : Cq 1, ◦ − D2(q 1), ◦ − C : 4, ◦ q+√2q+1 Cq √2q+1 : 4, ◦ − r Sz(q′), where q =(q′) for r a prime and q′ = 2. ◦ 6 Note: If 2f +1 is a prime, Sz(q) isa minimal simple group. ◦ 6.3

Octonion algebras

Let F be a field such that char(F ) = 2. ◦ 6 Hamilton H(F ) = 1, i, j, k [1843] • h iF are obtained from F by adjoining three orthogonal √ 1’s, ◦ − such that i j = k, j k = i, k i = j. ◦ · · · H(F ) is a skew-field such that dim (H(F )) = 4. ◦ F Letting H(F )′ := i, j, k = 1 ⊥, • h iF h iF with respect to the natural symmetric form, ◦ we have dim (H(F )′)=3, ◦ F yielding Aut(H(F )) = Aut(H(F )′) = SO (F ) = PGL (F ). • ∼ 3 ∼ 2

Cayley octonions O(F ) [Cayley, Graves, 1845/1843] • are obtained from F by adjoining seven orthogonal √ 1’s ◦ − i ,...,i , where any triple [i ,i ,i ] ◦{ 0 6} t t+1 t+3 fulfills the multiplication rules of i, j, k H(F ). ◦ ∈ O(F ) is a non-associative algebra such that dim (O(F )) = 8. ◦ F Letting O(F )′ := i ,...,i = 1 ⊥, • h 0 6iF h iF with respect to the natural symmetric form, ◦ we have dim (O(F )′)=7. ◦ F Replacing by a suitable form yields a characteristic-free definition: • 6.4

Octonion algebras, II

Chevalley group • G2(F ) ∼= Aut(O(F )) = Aut(O(F )′) < SO7(F ) The geometric approach yields, for example, ◦ G (q) = q6(q6 1)(q2 1); | 2 | − − G (F ) has a 7-dimensional natural representation, ◦ 2 while the adjoint representation has dimension 14. ◦ Exception to simplicity: G (2) = PSU (3): 2 ◦ 2 ∼ 3

Small 2G (32f+1)

Steinberg triality group G (q) < 3D (q3) < PΩ+(q3): • 2 4 8 automorphism group of twisted octonions. ◦ Note: 3D (q3)

Albert algebras

Let F be a finite field such that char(F ) 2, 3 . ◦ 6∈ { } Jordan product A B := 1(AB+BA) on an associative algebra • ◦ 2 is commutative, non-associative, and fufills the Jordan identity ◦ ((A A) B) A =(A A) (B A). ◦ ◦ ◦ ◦ ◦ ◦ A Jordan algebra is a commutative, non-associative algebra ◦ fullfing the Jordan identity.

Any simple Jordan F -algebra arises from an associative F -algebra, • except the Albert algebra • 3 3 tr A(F ) := A O(F ) × ; A = A , { ∈ } where : O(F ) O(F ) denotes octonion conjugation; ◦ → we have dim (A(F )) = 27. ◦ F

Letting A(F )′ := A A(F ); Tr(A)=0 = E ⊥, • { ∈ } h 3i with respect to the natural symmetric form, ◦ we have dim (A(F )′) = 26. ◦ F Replacing by a suitable form yields a characteristic-free definition: • 6.6

Albert algebras, II

Chevalley group F (q) = Aut(A(F )): • 4 ∼ q has a 26-dimensional natural representation, ◦ while the adjoint representation has dimension 52. ◦

Large Ree group 2F (22f+1)

Chevalley group E (q): [Dickson, 1901] • 6 leaves invariant a cubic ‘’ form on A(F ); ◦ q E (q) has a 27-dimensional natural representation, ◦ 6 while the adjoint representation has dimension 78. ◦

Steinberg group 2E (q2)

Golay codes

A S(t,k,v) on the set 1,...,v • { } is a set of k-, called blocks, such that ◦ any of size t is contained in precisely one block. ◦ Hence there are S(t,k,v) = v / k blocks. ◦ | | t t   Example: The finite of order q • is a Steiner system S(2, q +1, q2 + q + 1), ◦ the blocks being the projective lines. ◦ Theorem: There is a unique Steiner system S(5, 8, 24). • Existence: Three successive one-point extensions of S(2, 5, 21) ◦ coming from the projective plane of order 4 [Witt, 1938]; ◦ or: the blocks are the 759 words of weight 8 of the ◦ self-dual extended binary Golay [24, 12, 8] -code < F24. ◦ 2 G24 2 Words of weight 8 are called octads [Todd, 1966]. • Computational combinatorial tool: [Curtis, 1976] ◦ (MOG) ◦ Weight enumerator T 24 +759 T 16 +2576 T 12 +759 T 8 +1, • · · · the 2576 words of weight 12 are called dodecads. ◦ 7.2

Golay codes, II

Given a dodecad, • S(5, 8, 24) induces a Steiner system S(5, 6, 12) on it, ◦ being unique isomorphism, ◦ having 132 blocks, called hexads. ◦ Attaching signs, the blocks yield the words of weight 6 of the • self-dual extended ternary Golay [12, 6, 6] -code < F12; ◦ 3 G12 3 weight enumerator 2 (12 T 12 + 220 T 9 + 132 T 6 + 1). ◦ · · · ·

Any word of weight 4 determines a in the • Golay cocode (Todd ) F24/ , ◦ 2 G24 where 6 mutually disjoint words determine the same coset. ◦ Hence any word of weight 4 yields a sextet, ◦ a partition of 1,..., 24 into 6 subsets of size 4, ◦ { } the union of any two of which is an octad; ◦ there are 1 24 = 1771 sextets. • 6 · 4  7.3

Mathieu groups [1861/1873]

Mathieu group M := Aut(S(5, 8, 24)) = Aut( ), • 24 ∼ G24 acts 5-transitively on 1,..., 24 : ◦ { } M := Stab (1) = Aut( ), • 23 M24 ∼ G23 where < F23 is the perfect binary Golay [23, 12, 7] -code; ◦ G23 2 2 Mathieu group M := Stab (1, 2); • 22 M24 M := Stab (1, 2, 3) = PSL (4), in natural 2-transitive action. ◦ 21 M24 ∼ 3 M = 24 23 22 21 20 48=210 33 5 7 11 23 •| 24| · · · · · · · · · · Simplicity of M : Apply Iwasawa’s Criterion • 24 to the transitive action on the sextets, with stabiliser 26 : (3. ). ◦ S6

M acts transitive on the dodecads, with point stabiliser ◦ 24 Mathieu group M = Aut(S(5, 6, 12)), Aut( ) = 2.M ; • 12 ∼ G12 ∼ 12 M24 6 3 M = | | = 95040 = 2 3 5 11. ◦| 12| 2576 · · · M acts sharply 5-transivitely on 1,..., 12 : ◦ 12 { } Mathieu group M := Stab (1), Aut( ) = 2 M , • 11 M12 G11 ∼ × 11 where < F11 is the perfect ternary Golay [11, 6, 5] -code; ◦ G11 3 3 M := Stab (1, 2) = .2, • 10 M12 ∼ A6 where Aut( ) = .22 and = .2 = PGL (9). ◦ A6 ∼ A6 S6 6∼ A6 6∼ 2 7.4

Leech lattice

212 : M afforded by the Golay code , • 24 G24 acts monomially on ◦ : [Leech, Witt, 1967/1940] • L the set of all x := [x ,...,x ] Z24 such that ◦ 1 24 ∈ x 1 24 x m (mod 2), for some m, ◦ i ≡ 4 i=1 i ≡ and i;Px k (mod 4) , for each k; ◦ { i ≡ } ∈ G24 with scalar product x, y := 1 24 x y Z. ◦ h i 8 · i=1 i i ∈ P Theorem: is the unique unimodular even lattice in R24 • L without roots, that is vectors of norm 2. ◦ := x ; x,x = n , for n 2N . •Ln { ∈L h i } ∈ 0 n Weight Θ := n N0 2n T Z[[T ]]: ◦ L ∈ |L |· ∈ P Θ = 1+ 196560 T 2 + 16773120 T 3 + 398034000 T 4 + L · · · ···

falls into classes of 48 mutually orthogonal vectors, •L8 called coordinate frames, ◦ hence there are 398034000 = 8292375 coordinate frames. ◦ 48 7.5

Conway groups [1969]

Conway group 2.Co := Aut( ) • 1 L Co = 1 8292375 212 M =221 39 54 72 11 13 23 ◦| 1| 2 · · ·| 24| · · · · · · Simplicity: Apply Iwasawa’s Criterion to ◦ the transitive action on coordinate frames, with stabiliser212 : M . ◦ 24 Smallest representation of dimension 24 is globally irreducible. ◦

Sublattice groups: 2.Co acts transitively on and . ◦ 1 L4 L6 Co := Stab (v) where v ; • 2 2.Co1 ∈L4 Conway group Co := Stab (w) where w . • 3 2.Co1 ∈L6

2.Co acts transitively on [v,v′] ; v + v′ , ◦ 1 { ∈L4 ×L4 ∈L6} McLaughlin group [1969] McL := Stab 1(v,v′). • 2.Co

2.Co acts transitively on [w, w′] ; w + w′ , ◦ 1 { ∈L6 ×L6 ∈L4} Higman-Sims group [1968] HS := Stab 1(w, w′). • 2.Co

Higman-Sims graph on z , z, w =3, z, w′ = 3 , • { ∈L4 h i h i − } vertices z, z′ being adjacent if z, z′ = 1, ◦ h i size n = 100, regular of valency k = 22; ◦ HS primitive of rank 3, with stabiliser M . ◦ 22 7.6

Suzuki chain

Let 3D Co [Atlas] • ∈ 1 have order 3 and centraliser C (3D) = 3 . ◦ Co1 ∼ ×A9 Letting ◦ > > > > > > > A9 A8 A7 A6 A5 A4 A3 A2 yields corresponding centralisers C ( ) ◦ Co1 Ai < < PSL (2) < PSU (3)

6.Suz < 2.Co induces a complex structure C on , • 1 L L such that 6.Suz = Aut( C) acts irreducibly. ◦ L 2. < H(R) binary icosahedral group [Hamilton, 1857], • A5 hence 2. 2.J < 2. 2.G (4) < 2.Co ◦ A5 ◦ 2 A4 ◦ 2 1 induces a quaternionic structure H on , ◦ L L such that 2.J < 2.G (4) = Aut( H) act irreducibly; ◦ 2 2 L note: this yields the exceptional 2-fold cover 2.G (4). ◦ 2 8.1

Fischer groups

A finite group G generated by • a conjugacy class of involutions, called 3-transpositions, ◦ such that the product of two transpositions has order at most 3, ◦ G′ = G′′, and any normal 2- or 3-subgroup is central, ◦ is called a 3-transposition group. ◦ Theorem: [Fischer, 1968/1971] • Let G be a 3-transposition group. Then G/Z(G) is isomorphic to: ; PSU (22), Sp (2), GOǫ (2); PΩǫ (3) : 2,Ω (3), SO (3); ◦Sn n 2n 2n 2n 2n+1 2n+1 or one of the Fischer groups Fi , Fi , Fi′ .2. ◦ 22 23 24 Key tool: Transposition graph ∆, • with vertices corresponding to the 3-transpositions, ◦ being adjacent if the 3-transpositions commute. ◦ Hence ∆ is regular, and G Aut(∆) is vertex-transitive. • ≤ Fi : n = 3510, k = 693, H = 2.PSU (2); ◦ 22 ∼ 6 Fi : n = 31671, k = 3510, H = 2.F i ; ◦ 23 ∼ 22 Fi′ .2: n = 306936, k = 31671, H = 2 Fi ◦ 24 ∼ × 23 Simplicity: Apply Iwasawa’s Criterion • to the above primitive rank 3 actions on the vertices of ∆. ◦ 8.2

The Monster

2 2 3-transposition groups 2 .PSU (2 ) < 2.F i

Fischer-Griess Monster (Friendly Giant) M [1973]: • a6-transposition group of order ◦ 808017424794512875886459904961710757005754368000000000 =246 320 59 76 112 133 17 19 23 29 31 41 47 59 71 · · · · · · · · · · · · · · 8 1053 ∼ · Smallest representation V has dimension 196883, ◦ carrying structure of non-associative Griess algebra [1980]. ◦ Construction needs a thorough analysis of and . ◦ L G24 The Leech lattice and Fischer groups are involved in M. ◦ 8.3

Monstrous Moonshine

McKay, Thompson [1979]: • Fourier expansion of the elliptic modular j-function ◦ 1 2 3 j 744 = q− + 196884 q + 21493760 q + 864299970 q + , − · · · ··· has coefficients being character degrees of M. ◦ Moonshine Conjectures: [Conway, Norton, 1979] • There is an infinite-dimensional graded M-module ◦ inducing a relation between conjugacy classes of M ◦ and modular functions of genus 0. ◦ Frenkel, Lepowsky, Meurman [1988]: • construction of moonshine module, ◦ using vertex operators from conformal field theory. ◦ Borcherds [1992]: • M-invariant vertex algebra on moonshine module, ◦ proving the Moonshine Conjectures. ◦ 8.4

How to construct a Monster? [Griess, Conway, 1980/1985]

G := C (2B) = 21+24.Co , • 1 M ∼ + 1 where 224 = /2 and G /Z(G ) = 224 : Co . ◦ ∼ L L 1 1 ∼ 1 Let G be the universal cover of G , then Z(G ) = V , • 1 1 1 ∼ 4 s t 1+24 givinge rise to groups G = G = G1 of shapee 2 .Co1, ◦ 1 6∼ 1 ∼ + with smallest faithful representations of dimension 212 and 24 212. ◦ · V = 98304 98280 299, where • |G1 ∼ ⊕ ⊕ 98304 = 4096 24=212 , acted on by Gs and 2.Co ; ◦ ∼ ⊗ ⊗L 1 1 24 24 2 Co2 = [1, 22, 1] uniserial, 2 : Co2 having linear character 1−, ◦ | 24 2 .Co1 98280 = (1−24 ) monomial action; ∼ 2 .Co2 ↑ 1 299 = S2( ) < , acted on by Co . ◦ ⊕ ∼ L L⊗L 1 Restrict to G >G = 21+24.(211 : M ) = 22+11+22.(2 M ), • 1 12 ∼ + 24 ∼ × 24 triality yields G

G1 98304 98280 299 ↓ ւ↓ց ւ↓ G12 98304 49152 48576 552 276 23 ↑ր ↑ տ↑↑ G2 147456 48576 828 23 8.5

Monstrous groups

C (2B) = 21+24.Co • M ∼ + 1

C (3A) = 3.F i′ • M ∼ 24

Baby Monster B: [Fischer, 1973] • a4-transposition group, arising as C (2A) = 2.B. ◦ M ∼ Smallest representation has dimension 4371, ◦ is irreducible except in characteristic 2, ◦ and contains a vector with stabiliser 2.2E (22): 2, yielding ◦ 6 smallest permutation representation on 13571955000 points ◦ [Leon, Sims, 1980]. ◦

Thompson group [1973] Th: • 3C M preimage of 3D with respect to 21+24.Co Co ◦ ∈ + 1 → 1 gives rise to C (3C) = 3 Th. ◦ M ∼ × C (2A) = 21+8. ◦ T h ∼ + A9 Smallest representation has dimension 248, ◦ is globally irreducible, ◦ and yields an embedding Th

Monstrous groups, II

Harada-Norton group [1973] HN: • 5A M preimage of 5B with respect to 21+24.Co Co ◦ ∈ + 1 → 1 gives rise to C (5A) = 5 HN. ◦ M ∼ × C (2B) = 21+8.( ).2 ◦ HN ∼ + A5 ×A5 Smallest representation has dimension 133 over Q[√5], ◦ is irreducible except in characteristic 2, ◦ and does not yield an embedding into E (5). ◦ 7

Held group [1968] He: • arises as C (7A) = 7 He. ◦ M ∼ × Any simple group having an involution centraliser 21+6 : PSL (2) ◦ 3 is isomorphic to PSL (2), M , or He. ◦ 5 24 8.7

Pariahs

There are just six sporadic groups not involved in M. ◦ Wilson: ‘The behaviour of these six groups is so bizarre that ◦ any attempt to describe them ends up looking like a disconnected sequence of unrelated facts — it is simply the nature of the subject.’

Janko group [1965] J : • 1 C (2A) = 2 ; ◦ J1 ∼ ×A5 J

Pariahs, II

O’Nan group [1973] ON: • Parker, Ryba [1988]: 3.ON < GL (F ) ◦ 452 7 Soicher [1990]: action on 122760 points ◦ Lyons group [1969] Ly: • C (2A) = 2. ◦ Ly ∼ A11 Meyer, Neutsch, Parker [1985]: Ly < GL (F ) ◦ 111 5 Janko group [1975] J : • 4 C (2A) = 21+12.(3.M : 2) ◦ J4 ∼ + 22 Norton, Parker, Thackray [1980]: J < GL (F ), ◦ 4 112 2 the original motivation to develop the MeatAxe. ◦

Computational techniques play an important role in the construction and analysis of the sporadic simple groups.