Suzuki sporadic group

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.
This article is about the sporadic simple group. For the infinite family of groups of Lie type found by Suzuki, see Suzuki groups.
Algebraic structure → Group theory
Group theory
Basic notions
Group homomorphisms
Finite groups
Classification of finite simple groups
Modular groups
  • PSL(2,Z)
  • SL(2,Z)
Topological and Lie groups
Infinite dimensional Lie group
  • O(∞)
  • SU(∞)
  • Sp(∞)
Algebraic groups

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)251,596,8001782
32 · U(4, 3) · 2319,595,52022,880
U(5, 2)13,685,76032,760
21+6 · U(4, 2)3,317,760135,135
35 : M111,924,560232,960
J2 : 21,209,600370,656
24+6 : 3A61,105,920405,405
(A4 × L3(4)) : 2483,840926,640
22+8 : (A5 × S3)368,6401,216,215
M12 : 2190,0802,358,720
32+4 : 2 · (A4 × 22) · 2139,9683,203,200
(A6 × A5) · 243,20010,378,368
(A6 × 32 : 4) · 225,92017,297,280
L3(3) : 211,23239,916,800
L2(25)7,80057,480,192
A72,520177,914,880

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 (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, 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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