Conway group Co1

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For general background and history of the Conway groups, see Conway group.

In mathematics, the Conway group Co1 is the largest of the three sporadic Conway 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 II25,1.

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[edit]

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).

Let the co-ordinates be arranged so that the 6 consecutive blocks of 4 form a sextet with respect to the Golay code.

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


{\mathbf 1/2}  \left ( \begin{matrix}
1 &  -1 & -1 & -1 \\
-1 &  1 & -1 & -1 \\
-1 & -1 &  1 & -1 \\
-1 & -1 & -1 & 1 \end{matrix} \right )

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.

Involutions in Co0 and Co1[edit]

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[edit]

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 II1,25 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[edit]

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.Sp4(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[edit]

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.An (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[edit]

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