Suzuki sporadic group

This article is about the sporadic simple group. For the infinite family of groups of Lie type found by Suzuki, see Suzuki groups.
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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

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 subgroups of Suz
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
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
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

  • 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