| Algebraic structure → Group theory Group theory |
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In the area of modern algebra known as group theory, the Suzuki groupSuz or Sz is a sporadic simple group of order
Contents
- 213 · 37 · 52 · 7 · 11 · 13 = 448345497600 ≈ 4×1011.
| Algebraic structure → Group theory Group theory |
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In the area of modern algebra known as group theory, the Suzuki groupSuz or Sz is a sporadic simple group of order
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.
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.
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.
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 |
In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order
808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000
= 246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71
≈ 8×1053.
In the mathematical classification of finite simple groups, there are 26 or 27 groups which do not fit into any infinite family. These are called the sporadic simple groups, or the sporadic finite groups, or just the sporadic groups.
In the area of modern algebra known as group theory, the Conway groups are the three sporadic simple groups Co1, Co2 and Co3 along with the related finite group Co0 introduced by (Conway 1968, 1969).
In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups M11, M12, M22, M23 and M24 introduced by Mathieu. They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objects. They are the first sporadic groups to be discovered.
In the area of modern algebra known as group theory, the Higman–Sims group HS is a sporadic simple group of order
In group theory, the Tits group2F4(2)′, named for Jacques Tits (French:[tits]), is a finite simple group of order
In the area of modern algebra known as group theory, the Lyons groupLy or Lyons-Sims groupLyS is a sporadic simple group of order
In the area of modern algebra known as group theory, the Held groupHe is a sporadic simple group of order
In the area of modern algebra known as group theory, the Rudvalis groupRu is a sporadic simple group of order
In mathematics, a Ree group is a group of Lie type over a finite field constructed by Ree from an exceptional automorphism of a Dynkin diagram that reverses the direction of the multiple bonds, generalizing the Suzuki groups found by Suzuki using a different method. They were the last of the infinite families of finite simple groups to be discovered.
In the area of modern algebra known as group theory, the Janko groupJ2 or the Hall-Janko groupHJ is a sporadic simple group of order
In the area of modern algebra known as group theory, the Mathieu groupM12 is a sporadic simple group of order
In the area of modern algebra known as group theory, the Mathieu groupM22 is a sporadic simple group of order
In the area of modern algebra known as group theory, the Mathieu groupM24 is a sporadic simple group of order
In the area of modern algebra known as group theory, the McLaughlin group McL is a sporadic simple group of order
In mathematics, II25,1 is the even 26-dimensional Lorentzian unimodular lattice. It has several unusual properties, arising from Conway's discovery that it has a norm zero Weyl vector. In particular it is closely related to the Leech lattice Λ, and has the Conway group Co1 at the top of its automorphism group.
In the area of modern algebra known as group theory, the Conway groupCo2 is a sporadic simple group of order
In the area of modern algebra known as group theory, the Conway group is a sporadic simple group of order
In the area of modern algebra known as group theory, the Conway groupCo1 is a sporadic simple group of order