Suzuki sporadic group
| Algebraic structure → Group theory Group theory |
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In the area of modern algebra known as group theory, the Suzuki group Suz or Sz is a sporadic simple group of order
- 448,345,497,600 = 213 · 37 · 52 · 7 · 11 · 13 ≈ 4×1011.
History
[edit]Suz is one of the 26 Sporadic groups and was discovered by Suzuki (1969) as a rank 3 permutation group on 1782 points with point stabilizer G2(4). It is not related to the Suzuki groups of Lie type. The Schur multiplier has order 6 and the outer automorphism group has order 2.
Complex Leech lattice
[edit]The 24-dimensional Leech lattice has a fixed-point-free automorphism of order 3. Identifying this with a complex cube root of 1 makes the Leech lattice into a 12 dimensional lattice over the Eisenstein integers, called the complex Leech lattice. The automorphism group of the complex Leech lattice is the universal cover 6 · Suz of the Suzuki group. This makes the group 6 · Suz · 2 into a maximal subgroup of Conway's group Co0 = 2 · Co1 of automorphisms of the Leech lattice, and shows that it has two complex irreducible representations of dimension 12. The group 6 · Suz acting on the complex Leech lattice is analogous to the group 2 · Co1 acting on the Leech lattice.
Suzuki chain
[edit]The Suzuki chain or Suzuki tower is the following tower of rank 3 permutation groups from (Suzuki 1969), each of which is the point stabilizer of the next.
- G2(2) = U(3, 3) · 2 has a rank 3 action on 36 = 1 + 14 + 21 points with point stabilizer PSL(3, 2) · 2
- J2 · 2 has a rank 3 action on 100 = 1 + 36 + 63 points with point stabilizer G2(2)
- G2(4) · 2 has a rank 3 action on 416 = 1 + 100 + 315 points with point stabilizer J2 · 2
- Suz · 2 has a rank 3 action on 1782 = 1 + 416 + 1365 points with point stabilizer G2(4) · 2
Maximal subgroups
[edit]Wilson (1983) found the 17 conjugacy classes of maximal subgroups of Suz as follows:
| No. | Structure | Order | Index | Comments |
|---|---|---|---|---|
| 1 | G2(4) | 251,596,800 = 212·33·52·7·13 |
1,782 = 2·34·11 |
|
| 2 | 32· U(4, 3) : 2'3 | 19,595,520 = 28·37·5·7 |
22,880 = 25·5·11·13 |
normalizer of a subgroup of order 3 (class 3A) |
| 3 | U(5, 2) | 13,685,760 = 210·35·5·11 |
32,760 = 23·32·5·7·13 |
|
| 4 | 21+6 – · U(4, 2) |
3,317,760 = 213·34·5 |
135,135 = 33·5·7·11·13 |
centralizer of an involution of class 2A |
| 5 | 35 : M11 | 1,924,560 = 24·37·5·11 |
232,960 = 29·5·7·13 |
|
| 6 | J2 : 2 | 1,209,600 = 28·33·52·7 |
370,656 = 25·3^4·11·13 |
the subgroup fixed by an outer involution of class 2C |
| 7 | 24+6 : 3A6 | 1,105,920 = 213·33·5 |
405,405 = 34·5·7·11·13 |
|
| 8 | (A4 × L3(4)) : 2 | 483,840 = 29·33·5·7 |
926,640 = 24·34·5·11·13 |
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| 9 | 22+8 : (A5 × S3) | 368,640 = 213·32·5 |
1,216,215 = 35·5·7·11·13 |
|
| 10 | M12 : 2 | 190,080 = 27·33·5·11 |
2,358,720 = 26·34·5·7·13 |
the subgroup fixed by an outer involution of class 2D |
| 11 | 32+4 : 2(A4 × 22).2 | 139,968 = 26·37 |
3,203,200 = 27·52·7·11·13 |
|
| 12 | (A6 × A5) · 2 | 43,200 = 26·33·52 |
10,378,368 = 27·3^4·7·11·13 |
|
| 13 | (A6 × 32 : 4) · 2 | 25,920 = 26·34·5 |
17,297,280 = 27·33·5·7·11·13 |
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| 14,15 | L3(3) : 2 | 11,232 = 25·33·13 |
39,916,800 = 28·34·5^2·7·11 |
two classes, fused by an outer automorphism |
| 16 | L2(25) | 7,800 = 23·3·52·13 |
57,480,192 = 210·36·7·11 |
|
| 17 | A7 | 2,520 = 23·32·5·7 |
177,914,880 = 210·35·5·11·13 |
References
[edit]- Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A.: "Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups." Oxford, England 1985.
- 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
- Suzuki, Michio (1969), "A simple group of order 448,345,497,600", in Brauer, R.; Sah, Chih-han (eds.), Theory of Finite Groups (Symposium, Harvard Univ., Cambridge, Mass., 1968), Benjamin, New York, pp. 113–119, MR 0241527
- Wilson, Robert A. (1983), "The complex Leech lattice and maximal subgroups of the Suzuki group", Journal of Algebra, 84 (1): 151–188, doi:10.1016/0021-8693(83)90074-1, ISSN 0021-8693, MR 0716777
- 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