In the area of modern algebra known as
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, the Suzuki group ''Suz'' or ''Sz'' is a
sporadic simple group
In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups.
A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. The ...
of
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
: 2
13 · 3
7 · 5
2 · 7 · 11 · 13 = 448345497600
: ≈ 4.
History
''Suz'' is one of the 26 Sporadic groups and was discovered by as a
rank 3 permutation group
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* H ...
on 1782 points with point stabilizer G
2(4). It is not related to the
Suzuki groups of Lie type. The
Schur multiplier
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations.
Examples and properties
The Schur multiplier \op ...
has order 6 and the
outer automorphism group In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a ...
has order 2.
Complex Leech lattice
The 24-dimensional
Leech lattice
In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space, which is one of the best models for the kissing number problem. It was discovered by . It may also have been discovered (but not published) by ...
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
In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are the complex numbers of the form
:z = a + b\omega ,
where and are integers and
:\omega = \ ...
, 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 Co
0 = 2 · Co
1 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 · Co
1 acting on the Leech lattice.
Suzuki chain
The Suzuki chain or Suzuki tower is the following tower of
rank 3 permutation group
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* H ...
s from , each of which is the point stabilizer of the next.
* ''G''
2(2) = ''U''(3, 3) · 2 has a rank 3 action on 36 = 1 + 14 + 21 points with point stabilizer PSL(3, 2) · 2
* ''J''
2 · 2 has a rank 3 action on 100 = 1 + 36 + 63 points with point stabilizer ''G''
2(2)
* ''G''
2(4) · 2 has a rank 3 action on 416 = 1 + 100 + 315 points with point stabilizer J
2 · 2
* Suz · 2 has a rank 3 action on 1782 = 1 + 416 + 1365 points with point stabilizer G
2(4) · 2
Maximal subgroups
found the 17 conjugacy classes of maximal subgroups of ''Suz'' as follows:
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.
*
*
*
*
External links
MathWorld: Suzuki group Atlas of Finite Group Representations: Suzuki group
{{DEFAULTSORT:Suzuki Sporadic Group
Sporadic groups