Suzuki sporadic group
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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 Suzuki group Suz or Sz is a sporadic simple group of order
- 213 · 37 · 52 · 7 · 11 · 13 = 448345497600
- ≈ 4×1011.
History
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
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
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
Wilson (1983) found the 17 conjugacy classes of maximal subgroups of Suz as follows:
| Maximal Subgroup | Order | Index |
|---|---|---|
| G2(4) | 251,596,800 | 1782 |
| 32 · U(4, 3) · 23 | 19,595,520 | 22,880 |
| U(5, 2) | 13,685,760 | 32,760 |
| 21+6 · U(4, 2) | 3,317,760 | 135,135 |
| 35 : M11 | 1,924,560 | 232,960 |
| J2 : 2 | 1,209,600 | 370,656 |
| 24+6 : 3A6 | 1,105,920 | 405,405 |
| (A4 × L3(4)) : 2 | 483,840 | 926,640 |
| 22+8 : (A5 × S3) | 368,640 | 1,216,215 |
| M12 : 2 | 190,080 | 2,358,720 |
| 32+4 : 2 · (A4 × 22) · 2 | 139,968 | 3,203,200 |
| (A6 × A5) · 2 | 43,200 | 10,378,368 |
| (A6 × 32 : 4) · 2 | 25,920 | 17,297,280 |
| L3(3) : 2 | 11,232 | 39,916,800 |
| L2(25) | 7,800 | 57,480,192 |
| A7 | 2,520 | 177,914,880 |
References
- 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, 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 716777
- Wilson, Robert A. (2009), The finite simple groups, Graduate Texts in Mathematics 251 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 05622792