Conway group Co1

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In the area of modern algebra known as group theory, the Conway group Co₁ is a sporadic simple group of order

   2²¹ · 3⁹ · 5⁴ · 7² · 11 · 13 · 23

= 4157776806543360000

≈ 4×10¹⁸.

History and properties

Co₁ is one of the 26 sporadic groups and was discovered by John Horton Conway in 1968. It is the largest of the three sporadic Conway groups and can be 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.

Co₀ has no matrices of determinant -1.

An inner product on the Leech lattice is defined as 1/8 the sum of the products of respective co-ordinates of the two multiplicand vectors. The norm of a vector is its inner product with itself. It is common to speak of the type of a Leech lattice vector: half the norm. This lattice has no vectors of type 1. The matrices of Co₀ are orthogonal, I. e. they leave the inner product invariant. The inverse is the transpose.

Monomial subgroup of Co₀, N ≈ 2¹²:M₂₄

Conway started his investigation of Co₀ with a subgroup often called N, a holomorph of the binary Golay code (as diagonal matrices) by the Mathieu group M₂₄ (as permutation matrices).

A standard representation, used throughout this article, of the (extended) binary Golay code arranges the 24 co-ordinates so that 6 consecutive blocks (tetrads) of 4 constitute a sextet.

The Leech lattice can easily be defined as the Z-module generated by the set Λ₂ of all vectors of type 2, consisting of

(4,4,0²²)

(2⁸,0¹⁶)

(-3,1²³)

and their images under N. Under N Λ₂ falls into 3 orbits of sizes 1104, 97152, and 98304.Then |Λ₂| = 196560 = 2⁴*3³*5*7*13. Conway strongly suspected Co₀ was transitive on Λ₂, and indeed he found a new matrix, not monomial and not an integer matrix.

Let η be the 4-by-4 matrix

Now let ζ be a block sum of 6 matrices: odd numbers each of η and its negative.^[1][2] ζ is a symmetric and orthogonal matrix, thus an involution. Some experimenting shows that it interchanges vectors between different orbits of N.

To compute |Co₀| it is best to consider Λ₄, the set of vectors of type 4. For any type 4 vector, there are exactly 48 type 4 vectors congruent to it modulo 2Λ, falling into 24 orthogonal pairs {v,–v}. A set of 48 such vectors is called a frame or cross. N has as an orbit the frame of vectors with just one non-zero component, equal ±8. The subgroup fixing a given frame is a conjugate of N. The group 2¹² is isomorphic to the Golay code and acts as sign changes on vectors of the frame, while M₂₄ permutes the 24 pairs of the frame. Co₀ can be shown to be transitive on Λ₄. Conway multiplied the order 2¹²|M₂₄| of N by the number of frames, the latter being equal to the quotient |Λ₄|/48 = 8,252,375 = 3⁶*5³*7*13. That product is the order of any subgroup of Co₀ that properly contains N; hence N is a maximal subgroup of Co₀ and contains 2-Sylow subgroups of Co₀. N also is the subgroup in Co₀ of all matrices with integer components.

Since Λ includes vectors of the shape (±8,0²³), Co₀ consists of rational matrices whose denominators are all divisors of 8.

Involutions in Co₀ and Co₁

Any involution in Co₀ can be shown to be conjugate to an element of the Golay code. Co₀ has 4 conjugacy classes of involutions; these collapse to 2 in Co₁, but there are 4-elements in Co₀ that correspond to a third class of involutions in Co₁.

A permutation matrix of shape 2¹² can be shown to be conjugate to a dodecad. Its centralizer has the form 2¹²:M₁₂ and has conjugates inside the monomial subgroup.

A permutation matrix of shape 2⁸1⁸ can be shown to be conjugate to an octad. This and its negative have a common centralizer of the form (2¹⁺⁸x2).O₈⁺(2), a subgroup maximal in Co₀.

Representations

The smallest non-trivial representation of Co₀ over any field is the 24-dimensional one coming from the Leech lattice, and this is faithful over fields of characteristic other than 2.

The smallest faithful permutation representation of Co₁ is on the 98280 pairs {v,–v} of norm 4 vectors.

The centralizer of an involution of type 2B in the monster group is of the form 2¹⁺²⁴Co₁.

The Dynkin diagram of the even Lorentzian unimodular lattice II_1,25 is isometric to the (affine) Leech lattice Λ, so the group of diagram automorphisms is split extension Λ,Co₀ of affine isometries of the Leech lattice.

Maximal subgroups of Co₁

Wilson (1983) classified the maximal subgroups, though there were some errors in this list, corrected by Wilson (1988). Co₁ has 22 conjugacy classes of maximal subgroups as follows.

• Co₂ The lift to Aut(Λ) fixes a type 2 (norm 4) vector.
• 3.Suz:2 The lift to Aut(Λ) fixes a complex structure or changes it to the complex conjugate structure. Top of Suzuki chain; see below.
• 2¹¹:M₂₄ The lift to Aut(Λ) fixes a frame; see above.
• Co₃ The lift to Aut(Λ) fixes a type 3 (norm 6) vector.
• 2¹⁺⁸.O₈⁺(2)
• U₆(2):S₃
• (A₄ × G₂(4)):2 in Suzuki chain.
• 2²⁺¹²:(A₈ × S₃)
• 2⁴⁺¹².(S₃ × 3.S₆)
• 3².U₄(3).D₈
• 3⁶:2.M₁₂ (holomorph of ternary Golay code)
• (A₅ × J₂):2 in Suzuki chain
• 3¹⁺⁴:2.PSp₄(3).2
• (A₆ × U₃(3)).2 in Suzuki chain
• 3³⁺⁴:2.(S₄ × S₄)
• A₉ × S₃ in Suzuki chain
• (A₇ × L₂(7)):2 in Suzuki chain
• (D₁₀ × (A₅ × A₅).2).2
• 5¹⁺²:GL₂(5)
• 5³:(4 × A₅).2
• 7²:(3 × 2.S₄)
• 5²:2A₅

Suzuki chain of product groups

Co₀ (as well as its quotient Co₁) has 4 conjugacy classes of elements of order 3. In M₂₄ an element of shape 3⁸ generates a group normal in a copy of S₃, which commutes with a simple subgroup of order 168. The product PSL(2,7) x S₃ permutes the octads of a trio. In Co₀ this normalizer is expanded to a maximal subgroup of the form 2.A₉ x S₃, where 2.A₉ is the double cover of the alternating group A₉.

John Thompson pointed out it would be fruitful to investigate the normalizers of smaller subgroups of the form 2.A_n (Conway 1971, p.242). 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, the only one of this chain not maximal in Co₀. Next there is the subgroup (2.A₇ x PSL₂(7)):2. 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) being an exceptional group of Lie type.

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

References

1. ↑ Griess, p. 97.
2. ↑ Thomas Thompson, pp. 148-152.

• 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, MR 0237634 
• Brauer, R.; Sah, Chih-han, eds. (1969), Theory of finite groups: A symposium, W. A. Benjamin, Inc., New York-Amsterdam, MR 0240186 
• Conway, John Horton (1969), "A group of order 8,315,553,613,086,720,000", The Bulletin of the London Mathematical Society 1: 79–88, doi:10.1112/blms/1.1.79, ISSN 0024-6093, MR 0248216 
• Conway, John Horton (1971), "Three lectures on exceptional groups", in Powell, M. B.; Higman, Graham, Finite simple groups, Proceedings of an Instructional Conference organized by the London Mathematical Society (a NATO Advanced Study Institute), Oxford, September 1969., Boston, MA: Academic Press, pp. 215–247, ISBN 978-0-12-563850-0, MR 0338152  Reprinted in Conway & Sloane (1999, 267-298)
• Conway, John Horton; Sloane, Neil J. A. (1999), Sphere Packings, Lattices and Groups, Grundlehren der Mathematischen Wissenschaften 290 (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98585-5, MR 0920369 
• Thompson, Thomas M. (1983), From error-correcting codes through sphere packings to simple groups, Carus Mathematical Monographs 21, Mathematical Association of America, ISBN 978-0-88385-023-7, MR 749038 
• Conway, John Horton; Parker, Richard A.; Norton, Simon P.; Curtis, R. T.; Wilson, Robert A. (1985), Atlas of finite groups, Oxford University Press, ISBN 978-0-19-853199-9, MR 827219 
• Griess, Robert L. Jr. (1998), Twelve sporadic groups, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-62778-4, MR 1707296 
• Wilson, Robert A. (1983), "The maximal subgroups of Conway's group Co₁", Journal of Algebra 85 (1): 144–165, doi:10.1016/0021-8693(83)90122-9, ISSN 0021-8693, MR 723071 
• Wilson, Robert A. (1988), "On the 3-local subgroups of Conway's group Co₁", Journal of Algebra 113 (1): 261–262, doi:10.1016/0021-8693(88)90192-5, ISSN 0021-8693, MR 928064 
• Wilson, Robert A. (2009), The finite simple groups., Graduate Texts in Mathematics 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 05622792 

External links

• MathWorld: Conway Groups
• Atlas of Finite Group Representations: Co₁ version 2
• Atlas of Finite Group Representations: Co₁ version 3