Conway group Co3
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Modular groups
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Infinite dimensional Lie group
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In the area of modern algebra known as group theory, the Conway group Co3 is a sporadic simple group of order
- 210 · 37 · 53 · 7 · 11 · 23
- = 495766656000
- ≈ 5×1011.
History and properties
Co3 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 3, thus length √ 6. It is thus a subgroup of Co0. It is isomorphic to a subgroup of Co1. The direct product 2xCo3 is maximal in Co0.
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.
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.
Maximal subgroups
Finkelstein (1973) found the 14 conjugacy classes of maximal subgroups of Co3 as follows:
- McL:2 – can transpose type 2 points of conserved 2-2-3 triangle. Co3 has a doubly transitive permutation representation on 276 type 2-2-3 triangles containing a fixed type 3 point.
- HS – fixes 2-3-3 triangle.
- U4(3).22
- M23
- 35:(2 × M11)
- 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
- A4 × S5
Generalized Monstrous Moonshine
In analogy to monstrous moonshine for the monster M, for Co3, the relevant McKay-Thompson series is where one can set the constant term a(0) = 24 ( A097340),
and η(τ) is the Dedekind eta function.
References
- 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, doi:10.1073/pnas.61.2.398, MR 0237634
- 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, 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 290 (3rd ed.), Berlin, New York: Springer-Verlag, 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: 633–683, doi:10.1112/plms/s3-29.4.633, ISSN 0024-6115, MR 0374248
- Finkelstein, Larry (1973), "The maximal subgroups of Conway's group C₃ and McLaughlin's group", Journal of Algebra 25: 58–89, doi:10.1016/0021-8693(73)90075-6, ISSN 0021-8693, MR 0346046
- Thompson, Thomas M. (1983), From error-correcting codes through sphere packings to simple groups, Carus Mathematical Monographs 21, Mathematical Association of America, ISBN 978-0-88385-023-7, MR 749038
- Conway, John Horton; Parker, Richard A.; Norton, Simon P.; Curtis, R. T.; Wilson, Robert A. (1985), Atlas of finite groups, Oxford University Press, ISBN 978-0-19-853199-9, MR 827219
- Griess, Robert L. Jr. (1998), Twelve sporadic groups, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-62778-4, MR 1707296
- Wilson, Robert A. (2009), The finite simple groups., Graduate Texts in Mathematics 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 05622792
External links
- MathWorld: Conway Groups
- Atlas of Finite Group Representations: Co3 version 2
- Atlas of Finite Group Representations: Co3 version 3