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.

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 subgroup, 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 norm 4 vector.
  • 3.Suz:2 The lift to Aut(Λ) fixes a complex structure or changes it to the complex conjugate structure. Suzuki chain; see below
  • 211:M24 The lift to Aut(Λ) fixes a frame; see below.
  • Co3 The lift to Aut(Λ) fixes a norm 6 vector.
  • 21+8.O8+(2)
  • U6(2):S3
  • (A4 × G2(4)):2 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 Suzuki chain
  • 31+4:2.Sp4(3).2
  • (A6 × U3(3)).2 Suzuki chain
  • 33+4:2.(S4 × S4)
  • A9 × S3 Suzuki chain
  • (A7 × L2(7)):2 Suzuki chain
  • (D10 × (A5 × A5).2).2
  • 51+2:GL2(5)
  • 53:(4 × A5).2
  • 72:(3 × 2.S4)
  • 52:2A5

Maximal subgroup N = 212M24 of Co0[edit]

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. For any norm 8 vector of the Leech lattice, there are exactly 48 norm 8 vectors congruent to it modulo 2, falling into 24 orthogonal pairs {v,–v}. A set of 48 such vectors is called a frame. The subgroup fixing a frame is a group N which is a split extension of the form 212.M24, where the 212 is isomorphic to the Golay code and acts as sign changes on vectors of the frame, while the M24 permutes the 24 pairs of the frame. 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, as the order 212|M24| of N times the number of frames, where the number of frames is the number of norm 8 vectors divided by 48.

Suzuki chain of product groups[edit]

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.

John Thompson pointed out that was 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, but it is not maximal in Co0. Next there is the subgroup (2.A7 x PSL2(7)):2. 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.

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

References[edit]