Conway group Co2
Algebraic structure → Group theory Group theory |
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In the area of modern algebra known as group theory, the Conway group Co2 is a sporadic simple group of order
- 42,305,421,312,000
- = 218 · 36 · 53 · 7 · 11 · 23
- ≈ 4×1013.
History and properties
[edit]Co2 is one of the 26 sporadic groups and was discovered by (Conway 1968, 1969) 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
[edit]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.
Feit (1974) 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 =
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
[edit]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.
Wilson (2009) found the 11 conjugacy classes of maximal subgroups of Co2 as follows:
No. | Structure | Order | Index | Comments |
---|---|---|---|---|
1 | Fi21:2 ≈ U6(2):2 | 18,393,661,440 = 216·36·5·7·11 | 2,300 = 22·52·23 | 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. |
2 | 210:M22:2 | 908,328,960 = 218·32·5·7·11 | 46,575 = 34·52·23 | has monomial representation described above; 210:M22 fixes a 2-2-4 triangle. |
3 | McL | 898,128,000 = 27·36·53·7·11 | 47,104 = 211·23 | fixes a 2-2-3 triangle |
4 | 21+8 +:Sp6(2) | 743,178,240 = 218·34·5·7 | 56,925 = 32·52·11·23 | centralizer of an involution of class 2A (trace -8) |
5 | HS:2 | 88,704,000 = 210·32·53·7·11 | 476,928 = 28·34·23 | fixes a 2-3-3 triangle or exchanges its type 3 vertices with sign change |
6 | (24 × 21+6 +).A8 | 41,287,680 = 217·32·5·7 | 1,024,650 = 2·34·52·11·23 | centralizer of an involution of class 2B |
7 | U4(3):D8 | 26,127,360 = 210·36·5·7 | 1,619,200 = 28·52·11·23 | |
8 | 24+10.(S5 × S3) | 11,796,480 = 218·32·5 | 3,586,275 = 34·52·7·11·23 | |
9 | M23 | 10,200,960 = 27·32·5·7·11·23 | 4,147,200 = 211·34·52 | fixes a 2-3-4 triangle |
10 | 31+4 +.21+4 –.S5 | 933,120 = 28·36·5 | 45,337,600 = 210·52·7·11·23 | normalizer of a subgroup of order 3 (class 3A) |
11 | 51+2 +:4S4 | 12,000 = 25·3·53 | 3,525,451,776 = 213·35·7·11·23 | normalizer of a subgroup of order 5 (class 5A) |
Conjugacy classes
[edit]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
[edit]- 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, Bibcode:1968PNAS...61..398C, doi:10.1073/pnas.61.2.398, MR 0237634, PMC 225171, PMID 16591697
- 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 (eds.), 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, vol. 290 (3rd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4757-2016-7, ISBN 978-0-387-98585-5, MR 0920369
- Feit, Walter (1974), "On integral representations of finite groups", Proceedings of the London Mathematical Society, Third Series, 29 (4): 633–683, doi:10.1112/plms/s3-29.4.633, ISSN 0024-6115, MR 0374248
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- Wilson, Robert A. (1983), "The maximal subgroups of Conway's group ·2", Journal of Algebra, 84 (1): 107–114, doi:10.1016/0021-8693(83)90069-8, ISSN 0021-8693, MR 0716772
- Wilson, Robert A. (2009), The finite simple groups., Graduate Texts in Mathematics 251, vol. 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 1203.20012
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