Conway group

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In mathematics, the Conway groups Co₁, Co₂, and Co₃ are three sporadic groups discovered by John Horton Conway. Thomas Thompson relates how John Leech about 1964 investigated close packings of spheres in Euclidean spaces of large dimension. One of Leech's discoveries was a lattice packing in 24-space, based on what came to be called the Leech lattice Λ. He wondered whether his lattice's symmetry group contained an interesting simple group, but felt he needed the help of someone better acquainted with group theory. He had to do much asking around because the mathematicians were pre-occupied with agendas of their own. John Conway agreed to look at the problem. John G. Thompson said he would be interested if he were given the order of the group. Conway expected to spend months or years on the problem, but found results in just a few sessions.

The largest of the Conway groups, Co₁, of order

4,157,776,806,543,360,000,

is obtained as the quotient of Co₀ (automorphism group of Λ) by its center, which consists of the scalar matrices ±1. The groups Co₂ (of order 42,305,421,312,000) and Co₃ (of order 495,766,656,000) consist of the automorphisms of Λ fixing a lattice vector of type 2 and a vector of type 3 respectively. (The type of a vector is half of its square norm, vˑv.) As the scalar −1 fixes no non-zero vector, we can regard these two groups as subgroups of Co₁.

Contents 1 Other sporadic groups 2 An important maximal subgroup of Co0 3 Maximal subgroups of Co1 4 Maximal subgroups of Co2 5 Maximal subgroups of Co3 6 References

[edit] Other sporadic groups

Conway and Thompson found that 4 recently discovered sporadic simple groups were isomorphic to subgroups or quotients of subgroups of Co₁.

Two of these (subgroups of Co₂ and Co₃) can be defined as pointwise stabilizers of triangles with vertices, of sum zero, of types 2 and 3. A 2-2-3 triangle is fixed by the McLaughlin group McL (order 898,128,000). A 2-3-3 triangle is fixed by the Higman-Sims group (order 44,352,000).

Two other sporadic groups can be defined as stabilizers of structures on the Leech lattice. Identifying R²⁴ with C¹² and Λ with

Z[e^2πi/3]¹²,

the resulting automorphism group, i.e., the group of Leech lattice automorphisms preserving the complex structure, when divided by the 6-element group of complex scalar matrices, gives the Suzuki group Suz (of order 448,345,497,600). Suz is the only sporadic proper subquotient of Co₁ that retains 13 as a prime factor. This group was discovered by Michio Suzuki in 1968.

A similar construction gives the Hall-Janko group J₂ (of order 604,800) as the quotient of the group of quaternionic automorphisms of Λ by the group ±1 of scalars.

The 7 simple groups described above comprise what Robert Griess calls the second generation of the Happy Family, which consists of the 20 sporadic simple groups found within the Monster group. Several of the 7 groups contain at least some of the 5 Mathieu groups, which comprise the first generation.

There was a conference on group theory held May 2-4, 1968, at Harvard University. Richard Brauer and Chih-Han Shah later published a book of its proceedings. It included important lectures on four groups of the second generation, but was a little too early to include the Conway groups. It has on the other hand been observed that if Conway had started a few years earlier, he could have discovered all 7 groups. Conway's work unified a number of rabbit trails into one system.

[edit] An important maximal subgroup of Co₀

Conway started his investigation with a subgroup called N. The Leech lattice is defined by use of the binary Golay code, whose automorphism group is the Mathieu group M₂₄. Let E be a multiplicative representation of this code, a group of diagonal 24-by-24 matrices whose diagonal elements equal 1 or -1. E is an abelian group of type 2¹². Define N as the holomorph E:M₂₄. Conway found that N is a maximal subgroup of Co₀ and contains 2-Sylow subgroups of Co₀. He used N to deduce the order of Co₀.

In the usual matrix representation of Co₀, N is the subgroup whose matrices have integer components. In the complement Co₁\N there is a generating matrix whose components are in the module Z/2. ^[1]

The negative of the identity is in E and commutes with every 24-by-24 matrix. Then Co₁ has a maximal subgroup with structure 2¹¹:M₂₄.

[edit] Maximal subgroups of Co₁

Co₁ has 22 conjugacy classes of maximal subgroups. The maximal subgroups of Co₁ are as follows.

• Co₂
• 3.Suz:2 (order divisible by 13)
• 2¹¹:M₂₄
• Co₃
• 2¹⁺⁸.O₈⁺(2)
• U₆(2):S₃
• (A₄ × G₂(4)):2 (order divisible by 13)
• 2²⁺¹²:(A₈ × S₃)
• 2⁴⁺¹².(S₃ × 3.S₆)
• 3².U₄(3).D₈
• 3⁶:2.M₁₂ (holomorph of ternary Golay code)
• (A₅ × J₂):2
• 3¹⁺⁴:2.Sp₄(3).2
• (A₆ × U₃(3)).2
• 3³⁺⁴:2.(S₄ × S₄)
• A₉ × S₃
• (A₇ × L₂(7)):2 (order divisible by 49)
• (D₁₀ × (A₅ × A₅).2).2
• 5¹⁺²:GL₂(5) (contains Sylow 5-subgroups of Co₁)
• 5³:(4 × A₅).2 (contains Sylow 5-subgroups of Co₁)
• 7²:(3 × 2.S₄) (order divisible by 49)
• 5²:2A₅

Co₁ contains non-abelian simple groups of some 35 isomorphism types, as subgroups or as quotients of subgroups.

[edit] Maximal subgroups of Co₂

There are 11 conjugacy classes of maximal subgroups.

• U₆(2):2
• 2¹⁰:M₂₂:2
• McL (fixing 2-2-3 triangle)
• 2¹⁺⁸:Sp₆(2)
• HS:2 (can transpose type 3 vertices of conserved 2-3-3 triangle)
• (2⁴ × 2¹⁺⁶).A₈
• U₄(3):D₈
• 2⁴⁺¹⁰.(S₅ × S₃)
• M₂₃
• 3¹⁺⁴.2¹⁺⁴.S₅
• 5¹⁺²:4S₄

[edit] Maximal subgroups of Co₃

There are 14 conjugacy classes of maximal subgroups. Co₃ is doubly transitive on 276 type 2-2-3 triangles containing a fixed type 3 point.

• McL:2 - can transpose type 2 points of conserved 2-2-3 triangle
• HS - fixes 2-3-3 triangle
• U₄(3).2²
• M₂₃
• 3⁵:(2 × M₁₁)
• 2.Sp₆(2) - centralizer of involution class 2A, which moves 240 of the 276 type 2-2-3 triangles
• U₃(5):S₃
• 3¹⁺⁴:4S₆
• 2⁴.A₈
• PSL(3,4):(2 × S₃)
• 2 × M₁₂ - centralizer of involution class 2B, which moves 264 of the 276 type 2-2-3 triangles
• [2¹⁰.3³]
• S₃ × PSL(2,8):3
• A₄ × S₅

[edit] References

1. ^ Thomas Thompson (1983), p. 152.

• Conway, J. H.: A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple groups, Proc. Nat. Acad. Sci. U.S.A. 61 (1968), 398-400.
• Richard Brauer and Chih-Han Shah, Theory of Finite Groups: A Symposium, W. A. Benjamin (1969)
• Conway, J. H.: Three lectures on exceptional groups, in Finite Simple Groups, M. B. Powell and G. Higman (editors), Academic Press, (1971), 215-247. Reprinted in J. H. Conway & N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer (1988), 267-298.
• Thompson, Thomas M.: From Error Correcting Codes through Sphere Packings to Simple Groups, Carus Mathematical Monographs, Mathematical Association of America (1983).
• Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A., Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray. Eynsham: Oxford University Press (1985), ISBN 0-19-853199-0
• Griess, Robert L.: Twelve Sporadic Groups, Springer-Verlag (1998).
• Atlas of Finite Group Representations: Co₁ version 2
• Atlas of Finite Group Representations: Co₁ version 3