Английская Википедия:Conway group Co2
In the area of modern algebra known as group theory, the Conway group Co2 is a sporadic simple group of order
History and properties
Co2 is one of the 26 sporadic groups and was discovered by Шаблон:Harvs as the group of automorphisms of the Leech lattice Λ fixing a lattice vector of type 2. It is thus a subgroup of Co0. It is isomorphic to a subgroup of Co1. The direct product 2×Co2 is maximal in Co0.
The Schur multiplier and the outer automorphism group are both trivial.
Representations
Co2 acts as a rank 3 permutation group on 2300 points. These points can be identified with planar hexagons in the Leech lattice having 6 type 2 vertices.
Co2 acts on the 23-dimensional even integral lattice with no roots of determinant 4, given as a sublattice of the Leech lattice orthogonal to a norm 4 vector. Over the field with 2 elements it has a 22-dimensional faithful representation; this is the smallest faithful representation over any field.
Шаблон:Harvtxt showed that if a finite group has an absolutely irreducible faithful rational representation of dimension 23 and has no subgroups of index 23 or 24 then it is contained in either Z/2Z × Co2 or Z/2Z × Co3.
The Mathieu group M23 is isomorphic to a maximal subgroup of Co2 and one representation, in permutation matrices, fixes the type 2 vector u = (-3,123). A block sum ζ of the involution η =
- <math>
{\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 ) </math> and 5 copies of -η also fixes the same vector. Hence Co2 has a convenient matrix representation inside the standard representation of Co0. The trace of ζ is -8, while the involutions in M23 have trace 8.
A 24-dimensional block sum of η and -η is in Co0 if and only if the number of copies of η is odd.
Another representation fixes the vector v = (4,-4,022). A monomial and maximal subgroup includes a representation of M22:2, where any α interchanging the first 2 co-ordinates restores v by then negating the vector. Also included are diagonal involutions corresponding to octads (trace 8), 16-sets (trace -8), and dodecads (trace 0). It can be shown that Co2 has just 3 conjugacy classes of involutions. η leaves (4,-4,0,0) unchanged; the block sum ζ provides a non-monomial generator completing this representation of Co2.
There is an alternate way to construct the stabilizer of v. Now u and u+v = (1,-3,122) are vertices of a 2-2-2 triangle (vide infra). Then u, u+v, v, and their negatives form a coplanar hexagon fixed by ζ and M22; these generate a group Fi21 ≈ U6(2). α (vide supra) extends this to Fi21:2, which is maximal in Co2. Lastly, Co0 is transitive on type 2 points, so that a 23-cycle fixing u has a conjugate fixing v, and the generation is completed.
Maximal subgroups
Some maximal subgroups fix or reflect 2-dimensional sublattices of the Leech lattice. It is usual to define these planes by h-k-l triangles: triangles including the origin as a vertex, with edges (differences of vertices) being vectors of types h, k, and l.
Шаблон:Harvtxt found the 11 conjugacy classes of maximal subgroups of Co2 as follows:
- Fi21:2 ≈ U6(2):2 - symmetry/reflection group of coplanar hexagon of 6 type 2 points. Fixes one hexagon in a rank 3 permutation representation of Co2 on 2300 such hexagons. Under this subgroup the hexagons are split into orbits of 1, 891, and 1408. Fi21 fixes a 2-2-2 triangle defining the plane.
- 210:M22:2 has monomial representation described above; 210:M22 fixes a 2-2-4 triangle.
- McL fixes a 2-2-3 triangle.
- 21+8:Sp6(2) - centralizer of involution class 2A (trace -8)
- HS:2 fixes a 2-3-3 triangle or exchanges its type 3 vertices with sign change.
- (24 × 21+6).A8
- U4(3):D8
- 24+10.(S5 × S3)
- M23 fixes a 2-3-4 triangle.
- 31+4.21+4.S5
- 51+2:4S4
Conjugacy classes
Traces of matrices in a standard 24-dimensional representation of Co2 are shown.[1] The names of conjugacy classes are taken from the Atlas of Finite Group Representations. [2]
Centralizers of unknown structure are indicated with brackets.
| Class | Order of centralizer | Centralizer | Size of class | Trace | |
|---|---|---|---|---|---|
| 1A | all Co2 | 1 | 24 | ||
| 2A | 743,178,240 | 21+8:Sp6(2) | 32·52·11·23 | -8 | |
| 2B | 41,287,680 | 21+4:24.A8 | 2·34·5211·23 | 8 | |
| 2C | 1,474,560 | 210.A6.22 | 23·34·52·7·11·23 | 0 | |
| 3A | 466,560 | 31+421+4A5 | 211·52·7·11·23 | -3 | |
| 3B | 155,520 | 3×U4(2).2 | 211·3·52·7·11·23 | 6 | |
| 4A | 3,096,576 | 4.26.U3(3).2 | 24·33·53·11·23 | 8 | |
| 4B | 122,880 | [210]S5 | 25·35·52·7·11·23 | -4 | |
| 4C | 73,728 | [213.32] | 25·34·53·7·11·23 | 4 | |
| 4D | 49,152 | [214.3] | 24·35·53·7·11·23 | 0 | |
| 4E | 6,144 | [211.3] | 27·35·53·7·11·23 | 4 | |
| 4F | 6,144 | [211.3] | 27·35·53·7·11·23 | 0 | |
| 4G | 1,280 | [28.5] | 210·36·52·7·11·23 | 0 | |
| 5A | 3,000 | 51+22A4 | 215·35·7·11·23 | -1 | |
| 5B | 600 | 5×S5 | 215·35·5·7·11·23 | 4 | |
| 6A | 5,760 | 3.21+4A5 | 211·34·52·7·11·23 | 5 | |
| 6B | 5,184 | [26.34] | 212·32·53·7·11·23 | 1 | |
| 6C | 4,320 | 6×S6 | 213·33·52·7·11·23 | 4 | |
| 6D | 3,456 | [27.33] | 211·33·53·7·11·23 | -2 | |
| 6E | 576 | [26.32] | 212·34·53·7·11·23 | 2 | |
| 6F | 288 | [25.32] | 213·34·53·7·11·23 | 0 | |
| 7A | 56 | 7×D8 | 215·36·53·11·233 | 3 | |
| 8A | 768 | [28.3] | 210·35·53·7·11·23 | 0 | |
| 8B | 768 | [28.3] | 210·35·53·7·11·23 | -2 | |
| 8C | 512 | [29] | 29·36·53·7·11·23 | 4 | |
| 8D | 512 | [29] | 29·36·53·7·11·23 | 0 | |
| 8E | 256 | [28] | 210·36·53·7·11·23 | 2 | |
| 8F | 64 | [26] | 212·36·53·7·11·23 | 2 | |
| 9A | 54 | 9×S3 | 217·33·53·7·11·23 | 3 | |
| 10A | 120 | 5×2.A4 | 215·35·52·7·11·23 | 3 | |
| 10B | 60 | 10×S3 | 216·35·52·7·11·23 | 2 | |
| 10C | 40 | 5×D8 | 215·36·52·7·11·23 | 0 | |
| 11A | 11 | 11 | 218·36·53·7·23 | 2 | |
| 12A | 864 | [25.33] | 213·33·53·7·11·23 | -1 | |
| 12B | 288 | [25.32] | 213·34·53·7·11·23 | 1 | |
| 12C | 288 | [25.32] | 213·34·53·7·11·23 | 2 | |
| 12D | 288 | [25.32] | 213·34·53·7·11·23 | -2 | |
| 12E | 96 | [25.3] | 213·35·53·7·11·23 | 3 | |
| 12F | 96 | [25.3] | 213·35·53·7·11·23 | 2 | |
| 12G | 48 | [24.3] | 214·35·53·7·11·23 | 1 | |
| 12H | 48 | [24.3] | 214·35·53·7·11·23 | 0 | |
| 14A | 56 | 5×D8 | 215·36·53·11·23 | -1 | |
| 14B | 28 | 14×2 | 216·36·53·11·23 | 1 | power equivalent |
| 14C | 28 | 14×2 | 216·36·53·11·23 | 1 | |
| 15A | 30 | 30 | 217·35·52·7·11·23 | 1 | |
| 15B | 30 | 30 | 217·35·52·7·11·23 | 2 | power equivalent |
| 15C | 30 | 30 | 217·35·52·7·11·23 | 2 | |
| 16A | 32 | 16×2 | 213·36·53·7·11·23 | 2 | |
| 16B | 32 | 16×2 | 213·36·53·7·11·23 | 0 | |
| 18A | 18 | 18 | 217·34·53·7·11·23 | 1 | |
| 20A | 20 | 20 | 216·36·52·7·11·23 | 1 | |
| 20B | 20 | 20 | 216·36·52·7·11·23 | 0 | |
| 23A | 23 | 23 | 218·36·53·7·11 | 1 | power equivalent |
| 23B | 23 | 23 | 218·36·53·7·11 | 1 | |
| 24A | 24 | 24 | 215·35·53·7·11·23 | 0 | |
| 24B | 24 | 24 | 215·35·53·7·11·23 | 1 | |
| 28A | 28 | 28 | 216·36·53·11·23 | 1 | |
| 30A | 30 | 30 | 217·35·52·7·11·23 | -1 | |
| 30B | 30 | 30 | 217·35·52·7·11·23 | 0 | |
| 30C | 30 | 30 | 217·35·52·7·11·23 | 0 |
References
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation Reprinted in Шаблон:Harvtxt
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Specific
External links
- MathWorld: Conway Groups
- Atlas of Finite Group Representations: Co2 version 2
- Atlas of Finite Group Representations: Co2 version 3
