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 Rudvalis group ''Ru'' is a
sporadic simple group of
order
: 2
143
35
371329
: = 145926144000
: ≈ 1.
History
''Ru'' is one of the 26 sporadic groups and was found by and constructed by . Its
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 \oper ...
has order 2, and its
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 t ...
is trivial.
In 1982
Robert Griess
Robert Louis Griess, Jr. (born 1945, Savannah, Georgia) is a mathematician working on finite simple groups and vertex algebras. He is currently the John Griggs Thompson Distinguished University Professor of mathematics at University of Michigan.
...
showed that ''Ru'' cannot be a
subquotient of the
monster group
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
246320597611213317192329314147 ...
.
[Griess (1982)] Thus it is one of the 6 sporadic groups called the
pariahs.
Properties
The Rudvalis group acts as a rank 3 permutation group on 4060 points, with one point stabilizer being the
Ree group In mathematics, a Ree group is a group of Lie type over a finite field constructed by 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 ...
2''F''
4(2), the automorphism group of the
Tits group
In group theory,
the Tits group 2''F''4(2)′, named for Jacques Tits (), is a finite simple group of order
: 211 · 33 · 52 · 13 = 17,971,200.
It is sometimes considered a 27th sporadic group ...
. This representation implies a
strongly regular graph
In graph theory, a strongly regular graph (SRG) is defined as follows. Let be a regular graph with vertices and degree . is said to be strongly regular if there are also integers and such that:
* Every two adjacent vertices have comm ...
srg(4060, 2304, 1328, 1208). That is, each vertex has 2304 neighbors and 1755 non-neighbors, any two adjacent vertices have 1328 common neighbors, while any two non-adjacent ones have 1208 .
Its
double cover acts on a 28-dimensional lattice over the
Gaussian integer
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /ma ...
s. The lattice has 4×4060 minimal vectors; if minimal vectors are identified whenever one is 1, ''i'', –1, or –''i'' times another, then the 4060 equivalence classes can be identified with the points of the rank 3 permutation representation. Reducing this lattice modulo the
principal ideal
In mathematics, specifically ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where ...
:
gives an action of the Rudvalis group on a 28-dimensional vector space over the field
with 2 elements. Duncan (2006) used the 28-dimensional lattice to construct a
vertex operator algebra
In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications, vertex operator algebras have proven usef ...
acted on by the double cover.
characterized the Rudvalis group by the centralizer of a central involution. gave another characterization as part of their identification of the Rudvalis group as one of the
quasithin group In mathematics, a quasithin group is a finite simple group that resembles a group of Lie type of rank at most 2 over a field of characteristic 2. More precisely it is a finite simple group of characteristic 2 type and width 2. Here characteristic ...
s.
Maximal subgroups
found the 15 conjugacy classes of maximal subgroups of ''Ru'' as follows:
*
2F
4(2) =
2F
4(2)'.2
* 2
6.U
3(3).2
* (2
2 × Sz(8)):3
* 2
3+8:L
3(2)
* U
3(5):2
* 2
1+4+6.S
5
* PSL
2(25).2
2
* A
8
* PSL
2(29)
* 5
2:4.S
5
* 3.A
6.2
2
* 5
1+2:
5* L
2(13):2
* A
6.2
2
* 5:4 × A
5
References
*
*
*
*
*
*
*
*
*
*{{Citation , last1=Wilson , first1=Robert A. , title=The geometry and maximal subgroups of the simple groups of A. Rudvalis and J. Tits , doi=10.1112/plms/s3-48.3.533 , mr=735227 , year=1984 , journal=Proceedings of the London Mathematical Society , series=Third Series , issn=0024-6115 , volume=48 , issue=3 , pages=533–563
External links
MathWorld: Rudvalis GroupAtlas of Finite Group Representations: Rudvalis group
Sporadic groups