Misplaced Pages

Algebraic structure → Group theory Group theory
Basic notions Subgroup Normal subgroup Group action Quotient group (Semi-)direct product Direct sum Free product Wreath product Group homomorphisms kernel image simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable Glossary of group theory List of group theory topics
Finite groups Cyclic group Zn Symmetric group Sn Alternating group An Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group Schur multiplier Classification of finite simple groups cyclic alternating Lie type sporadic
Discrete groups Lattices Integers (${\displaystyle \mathbb {Z} }$) Free group Modular groups PSL(2, ${\displaystyle \mathbb {Z} }$) SL(2, ${\displaystyle \mathbb {Z} }$) Arithmetic group Lattice Hyperbolic group
Topological and Lie groups Solenoid Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞)
Algebraic groups Linear algebraic group Reductive group Abelian variety Elliptic curve
v t e

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

   495,766,656,000

= 2 · 3 · 5 · 7 · 11 · 23

≈ 5×10.

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 ${\displaystyle \Lambda }$ 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

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.

Walter 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 ${\displaystyle \mathbb {Z} /2\mathbb {Z} \times \mathrm {Co} _{2}}$ 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 ${\displaystyle \mathrm {Co} _{3}}$ as follows:

Maximal subgroups of Co₃

No.	Structure	Order	Index	Comments
1	McL:2	1,796,256,000 = 2·3·5·7·11	276 = 2·3·23	McL fixes a 2-2-3 triangle. The maximal subgroup also includes reflections of the triangle. ${\displaystyle \mathrm {Co} _{3}}$ has a doubly transitive permutation representation on 276 type 2-2-3 triangles having as an edge a type 3 vector fixed by ${\displaystyle \mathrm {Co} _{3}}$.
2	HS	44,352,000 = 2·3·5·7·11	11,178 = 2·3·23	fixes a 2-3-3 triangle
3	U4(3).2	13,063,680 = 2·3·5·7	37,950 = 2·3·5·11·23
4	M23	10,200,960 = 2·3·5·7·11·23	48,600 = 2·3·5	fixes a 2-3-4 triangle
5	3:(2 × M11)	3,849,120 = 2·3·5·11	128,800 = 2·5·7·23	fixes or reflects a 3-3-3 triangle
6	2Sp6(2)	2,903,040 = 2·3·5·7	170,775 = 3·5·11·23	centralizer of an involution of class 2A (trace 8), which moves 240 of the 276 type 2-2-3 triangles
7	U3(5):S3	756,000 = 2·3·5·7	655,776 = 2·3·11·23
8	3 +:4S6	699,840 = 2·3·5	708,400 = 2·5·7·11·23	normalizer of a subgroup of order 3 (class 3A)
9	2A8	322,560 = 2·3·5·7	1,536,975 = 3·5·11·23
10	PSL(3,4):(2 × S3)	241,920 = 2·3·5·7	2,049,300 = 2·3·5·11·23
11	2 × M12	190,080 = 2·3·5·11	2,608,200 = 2·3·5·7·23	centralizer of an involution of class 2B (trace 0), which moves 264 of the 276 type 2-2-3 triangles
12		27,648 = 2·3	17,931,375 = 3·5·7·11·23
13	S3 × PSL(2,8):3	9,072 = 2·3·7	54,648,000 = 2·3·5·11·23	normalizer of a subgroup of order 3 (class 3C, trace 0)
14	A4 × S5	1,440 = 2·3·5	344,282,400 = 2·3·5·7·11·23

Conjugacy classes

Traces of matrices in a standard 24-dimensional representation of Co₃ are shown. The names of conjugacy classes are taken from the Atlas of Finite Group Representations. The cycle structures listed act on the 276 2-2-3 triangles that share the fixed type 3 side.

Generalized Monstrous Moonshine

In analogy to monstrous moonshine for the monster M, for Co₃, the relevant McKay-Thompson series is ${\displaystyle T_{4A}(\tau )}$ 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

1. Conway et al. (1985)
2. "ATLAS: Conway group Co3".
3. "ATLAS: Conway group Co1".
4. "ATLAS: Co3 — Permutation representation on 276 points".

• 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
• 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, vol. 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, vol. 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 1203.20012

External links

• MathWorld: Conway Groups
• Atlas of Finite Group Representations: Co₃ version 2
• Atlas of Finite Group Representations: Co₃ version 3

• Sporadic groups
• John Horton Conway