Conway group

Group theory
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In mathematics, the Conway groups Co₁, Co₂, and Co₃ are three sporadic groups discovered by John Horton Conway.

The largest of the Conway groups, Co₁, of order

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

is obtained as the quotient of Co₀ (group of automorphisms of the Leech lattice Λ that fix the origin) by its center, which consists of the scalar matrices ±1. It also appears at the top of the automorphism group of the even 26-dimensional unimodular lattice II_25,1. 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, v·v.) As the scalar −1 fixes no non-zero vector, these two groups are isomorphic to subgroups of Co₁.

History

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.

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 Sah 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 unified 4 seemingly rather unrelated groups into one larger group.

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.

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.

The matrices of N have components that are integers. Since N is maximal in Co₀, N is the group of all integral matrices in Co₀.

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.

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

Maximal subgroups of Co₃

There are 14 conjugacy classes of maximal subgroups. Co₃ has a doubly transitive permutation representation 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

A chain of product groups

Co₀ (as well as its quotient Co₁) has 4 conjugacy classes of elements of order 3. One of these commutes with a double cover of the alternating group A₉. In fact the normalizer of that 3-element has the form 2.A₉ x S₃. This maximal subgroup reveals interesting features not found in the Mathieu groups. It has a simple subgroup of order 504, containing an element of order 9.

It was fruitful to investigate the normalizers of smaller subgroups of the form 2.A_n. Several other maximal subgroups of Co₀ are found in this way. Moreover, two sporadic groups appear in the resulting chain.

There is a subgroup 2.A₈ x S₄, but it is not maximal in Co₀. Next there is the subgroup (2.A₇ x PSL₂(7)):2, whose order is divisible by 49. This group and the rest of the chain are maximal in Co₀. Next comes (2.A₆ x SU₃(3)):2. The unitary group SU₃(3) (order 6048) possesses a graph of 36 vertices, in anticipation of the next subgroup. That subgroup is (2.A₅ o 2.HJ):2. The aforementioned graph expands to the Hall-Janko graph, with 100 vertices. The Hall-Janko group HJ makes its appearance here. Next comes (2.A₄ o 2.G₂(4)):2. G₂(4) is an exceptional group of Lie type. Its order is divisible by 13, fairly rare among subgroups of the Conway groups.

The chain ends with 6.Suz:2 (Suz=Suzuki group), which, as mentioned above, respects a complex representaion of the Leech Lattice.

References

• Conway, John Horton (1968), "A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple groups", Proceedings of the National Academy of Sciences of the United States of America 61 (2): 398–400, doi:10.1073/pnas.61.2.398, MR0237634 
• Richard Brauer and Chih-Han Sah, Theory of Finite Groups: A Symposium, W. A. Benjamin (1969)
• Conway, J. H.: A group of order 8,315,553,613,086,720,000. Bull. London Math. Soc. 1 (1969), 79-88, the first-ever article on the group .0
• 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. Jr. (1998), Twelve sporadic groups, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-62778-4, MR1707296 
• Atlas of Finite Group Representations: Co₁ version 2
• Atlas of Finite Group Representations: Co₁ version 3
• Robert A. Wilson, 'The Finite Simple Groups', Springer-Verlag (2009).