Английская Википедия:Conway group Co3
In the area of modern algebra known as group theory, the Conway group <math>\mathrm{Co}_3</math> is a sporadic simple group of order
History and properties
<math>\mathrm{Co}_3</math> is one of the 26 sporadic groups and was discovered by Шаблон:Harvs as the group of automorphisms of the Leech lattice <math>\Lambda</math> fixing a lattice vector of type 3, thus length Шаблон:Radic. It is thus a subgroup of <math>\mathrm{Co}_0</math>. It is isomorphic to a subgroup of <math>\mathrm{Co}_1</math>. The direct product <math>2\times \mathrm{Co}_3</math> is maximal in <math>\mathrm{Co}_0</math>.
The Schur multiplier and the outer automorphism group are both trivial.
Representations
Co3 acts on the unique 23-dimensional even lattice of determinant 4 with no roots, given by the orthogonal complement of a norm 4 vector of the Leech lattice. This gives 23-dimensional representations over any field; over fields of characteristic 2 or 3 this can be reduced to a 22-dimensional faithful representation.
Co3 has a doubly transitive permutation representation on 276 points.
Шаблон:Harvs 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 <math>\Z/2\Z \times \mathrm{Co}_2</math> or <math>\Z/2\Z \times \mathrm{Co}_3</math>.
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:Harvs found the 14 conjugacy classes of maximal subgroups of <math>\mathrm{Co}_3</math> as follows:
- McL:2 – McL fixes a 2-2-3 triangle. The maximal subgroup also includes reflections of the triangle. <math>\mathrm{Co}_3</math> has a doubly transitive permutation representation on 276 type 2-2-3 triangles having as an edge a type 3 vector fixed by <math>\mathrm{Co}_3</math>.
- HS – fixes a 2-3-3 triangle.
- U4(3).22
- M23 – fixes a 2-3-4 triangle.
- 35:(2 × M11) - fixes or reflects a 3-3-3 triangle.
- 2.Sp6(2) – centralizer of involution class 2A (trace 8), 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 (trace 0), which moves 264 of the 276 type 2-2-3 triangles
- [210.33]
- S3 × PSL(2,8):3 - normalizer of 3-subgroup generated by class 3C (trace 0) element
- A4 × S5
Conjugacy classes
Traces of matrices in a standard 24-dimensional representation of Co3 are shown.[1] The names of conjugacy classes are taken from the Atlas of Finite Group Representations.[2] [3] The cycle structures listed act on the 276 2-2-3 triangles that share the fixed type 3 side.[4]
Class | Order of centralizer | Size of class | Trace | Cycle type | |
---|---|---|---|---|---|
1A | all Co3 | 1 | 24 | ||
2A | 2,903,040 | 33·52·11·23 | 8 | 136,2120 | |
2B | 190,080 | 23·34·52·7·23 | 0 | 112,2132 | |
3A | 349,920 | 25·52·7·11·23 | -3 | 16,390 | |
3B | 29,160 | 27·3·52·7·11·23 | 6 | 115,387 | |
3C | 4,536 | 27·33·53·11·23 | 0 | 392 | |
4A | 23,040 | 2·35·52·7·11·23 | -4 | 116,210,460 | |
4B | 1,536 | 2·36·53·7·11·23 | 4 | 18,214,460 | |
5A | 1500 | 28·36·7·11·23 | -1 | 1,555 | |
5B | 300 | 28·36·5·7·11·23 | 4 | 16,554 | |
6A | 4,320 | 25·34·52·7·11·23 | 5 | 16,310,640 | |
6B | 1,296 | 26·33·53·7·11·23 | -1 | 23,312,639 | |
6C | 216 | 27·34·53·7·11·23 | 2 | 13,26,311,638 | |
6D | 108 | 28·34·53·7·11·23 | 0 | 13,26,33,642 | |
6E | 72 | 27·35·53·7·11·23 | 0 | 34,644 | |
7A | 42 | 29·36·53·11·23 | 3 | 13,739 | |
8A | 192 | 24·36·53·7·11·23 | 2 | 12,23,47,830 | |
8B | 192 | 24·36·53·7·11·23 | -2 | 16,2,47,830 | |
8C | 32 | 25·37·53·7·11·23 | 2 | 12,23,47,830 | |
9A | 162 | 29·33·53·7·11·23 | 0 | 32,930 | |
9B | 81 | 210·33·53·7·11·23 | 3 | 13,3,930 | |
10A | 60 | 28·36·52·7·11·23 | 3 | 1,57,1024 | |
10B | 20 | 28·37·52·7·11·23 | 0 | 12,22,52,1026 | |
11A | 22 | 29·37·53·7·23 | 2 | 1,1125 | power equivalent |
11B | 22 | 29·37·53·7·23 | 2 | 1,1125 | |
12A | 144 | 26·35·53·7·11·23 | -1 | 14,2,34,63,1220 | |
12B | 48 | 26·36·53·7·11·23 | 1 | 12,22,32,64,1220 | |
12C | 36 | 28·35·53·7·11·23 | 2 | 1,2,35,43,63,1219 | |
14A | 14 | 29·37·53·11·23 | 1 | 1,2,751417 | |
15A | 15 | 210·36·52·7·11·23 | 2 | 1,5,1518 | |
15B | 30 | 29·36·52·7·11·23 | 1 | 32,53,1517 | |
18A | 18 | 29·35·53·7·11·23 | 2 | 6,94,1813 | |
20A | 20 | 28·37·52·7·11·23 | 1 | 1,53,102,2012 | power equivalent |
20B | 20 | 28·37·52·7·11·23 | 1 | 1,53,102,2012 | |
21A | 21 | 210·36·53·11·23 | 0 | 3,2113 | |
22A | 22 | 29·37·53·7·23 | 0 | 1,11,2212 | power equivalent |
22B | 22 | 29·37·53·7·23 | 0 | 1,11,2212 | |
23A | 23 | 210·37·53·7·11 | 1 | 2312 | power equivalent |
23B | 23 | 210·37·53·7·11 | 1 | 2312 | |
24A | 24 | 27·36·53·7·11·23 | -1 | 124,6,1222410 | |
24B | 24 | 27·36·53·7·11·23 | 1 | 2,32,4,122,2410 | |
30A | 30 | 29·36·52·7·11·23 | 0 | 1,5,152,308 |
Generalized Monstrous Moonshine
In analogy to monstrous moonshine for the monster M, for Co3, the relevant McKay-Thompson series is <math>T_{4A}(\tau)</math> where one can set the constant term a(0) = 24 (Шаблон:OEIS2C),
- <math>\begin{align}j_{4A}(\tau)
&=T_{4A}(\tau)+24\\ &=\Big(\tfrac{\eta^2(2\tau)}{\eta(\tau)\,\eta(4\tau)} \Big)^{24} \\ &=\Big(\big(\tfrac{\eta(\tau)}{\eta(4\tau)}\big)^{4}+4^2 \big(\tfrac{\eta(4\tau)}{\eta(\tau)}\big)^{4}\Big)^2\\ &=\frac{1}{q} + 24+ 276q + 2048q^2 +11202q^3+49152q^4+\dots \end{align}</math>
and η(τ) is the Dedekind eta function.
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
External links
- MathWorld: Conway Groups
- Atlas of Finite Group Representations: Co3 version 2
- Atlas of Finite Group Representations: Co3 version 3
- ↑ Conway et al. (1985)
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite web