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 Conway group ''Co
1'' 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 d ...
: 2
213
95
47
2111323
: = 4157776806543360000
: ≈ 4.
History and properties
''Co
1'' is one of the 26 sporadic groups and was discovered by
John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branc ...
in 1968. It is the largest of the three sporadic Conway groups and can be obtained as the quotient of ''Co
0'' (
group of automorphisms of the
Leech lattice Λ that fix the origin) by its
center, which consists of the scalar matrices ±1. It also appears at the top of the automorphism group of the even 26-dimensional unimodular lattice
II25,1. Some rather cryptic comments in Witt's collected works suggest that he found the Leech lattice and possibly the order of its automorphism group in unpublished work in 1940.
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 ...
is trivial and 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 2.
Involutions
Co
0 has 4 conjugacy classes of involutions; these collapse to 2 in Co
1, but there are 4-elements in Co
0 that correspond to a third class of involutions in Co
1.
An image of a dodecad has a centralizer of type 2
11:M
12:2, which is contained in a maximal subgroup of type 2
11:M
24.
An image of an octad or 16-set has a centralizer of the form 2
1+8.O
8+(2), a maximal subgroup.
Representations
The smallest faithful permutation representation of Co
1 is on the 98280 pairs of norm 4 vectors.
There is a matrix representation of dimension 24 over the field
.
The centralizer of an involution of type 2B in 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
2463205976112133171923293141475 ...
is of the form 2
1+24Co
1.
The Dynkin diagram of the even Lorentzian
unimodular lattice
In geometry and mathematical group theory, a unimodular lattice is an integral lattice of determinant 1 or −1. For a lattice in ''n''-dimensional Euclidean space, this is equivalent to requiring that the volume of any fundam ...
II1,25 is isometric to the (affine) Leech lattice Λ, so the group of diagram automorphisms is split extension Λ,Co
0 of affine isometries of the Leech lattice.
Maximal subgroups
found the 22 conjugacy classes of maximal subgroups of ''Co
1'', though there were some errors in this list, corrected by .
*
Co2
*3.
Suz:2 The lift to Aut(Λ) = Co
0 fixes a complex structure or changes it to the complex conjugate structure. Also, top of
Suzuki chain.
*2
11:
M24 Image of monomial subgroup from Aut(Λ), that subgroup stabilizing the standard
frame of 48 vectors of form (±8,0
23) .
*
Co3
*2
1+8.O
8+(2) centralizer of involution class 2A (image of octad from Aut(Λ))
*
Fi21:S
3 ≈ U
6(2):S
3 The lift to Aut(Λ) is the symmetry group of a coplanar hexagon of 6
type 2 points.
*(A
4 × G
2(4)):2 in Suzuki chain.
*2
2+12:(A
8 × S
3)
*2
4+12.(S
3 × 3.S
6)
*3
2.U
4(3).D
8
*3
6:2.
M12 (holomorph of
ternary Golay code)
*(A
5 × J
2):2 in Suzuki chain
*3
1+4:2.PSp
4(3).2
*(A
6 × U
3(3)).2 in Suzuki chain
*3
3+4:2.(S
4 × S
4)
*A
9 × S
3 in Suzuki chain
*(A
7 × L
2(7)):2 in Suzuki chain
*(D
10 × (A
5 × A
5).2).2
*5
1+2:GL
2(5)
*5
3:(4 × A
5).2
*7
2:(3 × 2.S
4)
*5
2:2A
5
References
*
*
*
* Reprinted in
*
*
*
*
*
*
*
External links
MathWorld: Conway GroupsAtlas of Finite Group Representations: Co1version 2
Atlas of Finite Group Representations: Co1version 3
{{DEFAULTSORT:Conway Group Co1
Sporadic groups
John Horton Conway