ART

In the area of modern algebra known as group theory, the Conway group \( {\displaystyle \mathrm {Co} _{3}} \( is a sporadic simple group of order

210 · 37 · 53 · 7 · 11 · 23
= 495766656000
≈ 5×1011.


History and properties

\( {\displaystyle \mathrm {Co} _{3}} \) is one of the 26 sporadic groups and was discovered by John Horton Conway (1968, 1969) as the group of automorphisms of the Leech lattice \( \Lambda \) fixing a lattice vector of type 3, thus length √6. It is thus a subgroup of \( {\displaystyle \mathrm {Co} _{0}} \). It is isomorphic to a subgroup of \( {\displaystyle \mathrm {Co} _{1}} \). The direct product \( {\displaystyle 2\times \mathrm {Co} _{3}} \) is maximal in \( {\displaystyle \mathrm {Co} _{0}} \).

The Schur multiplier and the outer automorphism group are both trivial.

Representations

\( {\displaystyle \mathrm {Co} _{3}} \) 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.

\( {\displaystyle \mathrm {Co} _{3}} \) has a doubly transitive permutation representation on 276 points.

(txt) 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 \( {\displaystyle \mathbb {Z} /2\mathbb {Z} \times \mathrm {Co} _{2}} \) or \({\displaystyle \mathbb {Z} /2\mathbb {Z} \times \mathrm {Co} _{3}} \).

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.

Larry Finkelstein (1973) found the 14 conjugacy classes of maximal subgroups of C o 3 {\displaystyle \mathrm {Co} _{3}} {\displaystyle \mathrm {Co} _{3}} as follows:

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 T\( T_{{4A}}(\tau ) \) where one can set the constant term a(0) = 24 (OEIS: A097340),

\( {\displaystyle {\begin{aligned}j_{4A}(\tau )&=T_{4A}(\tau )+24\\&={\Big (}{\tfrac {\eta ^{2}(2\tau )}{\eta (\tau )\,\eta (4\tau )}}{\Big )}^{24}\\&={\Big (}{\big (}{\tfrac {\eta (\tau )}{\eta (4\tau )}}{\big )}^{4}+4^{2}{\big (}{\tfrac {\eta (4\tau )}{\eta (\tau )}}{\big )}^{4}{\Big )}^{2}\\&={\frac {1}{q}}+24+276q+2048q^{2}+11202q^{3}+49152q^{4}+\dots \end{aligned}}} \)

and η(τ) is the Dedekind eta function.
References

Conway et al. (1985)
http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Co3/#ccls
http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Co1/#ccls

http://brauer.maths.qmul.ac.uk/Atlas/v3/permrep/Co3G1-p276B0

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, 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, 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: 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 0749038
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 0827219
Griess, Robert L. Jr. (1998), Twelve sporadic groups, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-662-03516-0, 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 1203.20012

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