| Algebraic structure → Group theory Group theory |
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In the area of modern algebra known as group theory, the Conway groupCo1 is a sporadic simple group of order
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- 4,157,776,806,543,360,000
- = 221 ·39 ·54 ·72 ·11 ·13 ·23
- ≈ 4×1018.
| Algebraic structure → Group theory Group theory |
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In the area of modern algebra known as group theory, the Conway groupCo1 is a sporadic simple group of order
Co1 is one of the 26 sporadic groups and was discovered by John Horton Conway in 1968. It is the largest of the three sporadic Conway groups and can be obtained as the quotient of Co0 (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in Witt's collected works suggest that he found the Leech lattice and possibly the order of its automorphism group in unpublished work in 1940.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 is trivial and the Schur multiplier has order 2.
Co0 has 4 conjugacy classes of involutions; these collapse to 2 in Co1, but there are 4-elements in Co0 that correspond to a third class of involutions in Co1.
An image of a dodecad has a centralizer of type 211:M12:2, which is contained in a maximal subgroup of type 211:M24.
An image of an octad or 16-set has a centralizer of the form 21+8.O8+(2), a maximal subgroup.
The smallest faithful permutation representation of Co1 is on the 98280 pairs {v,–v} 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 is of the form 21+24Co1.
The Dynkin diagram of the even Lorentzian unimodular lattice II1,25 is isometric to the (affine) Leech lattice Λ, so the group of diagram automorphisms is split extension Λ,Co0 of affine isometries of the Leech lattice.
Wilson (1983) found the 22 conjugacy classes of maximal subgroups of Co1, though there were some errors in this list, corrected by Wilson (1988).
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 the area of modern algebra known as group theory, the Suzuki groupSuz or Sz is a sporadic simple group of order
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 Thompson groupTh 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 abstract algebra known as group theory, the O'Nan groupO'N or O'Nan–Sims group 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 the area of modern algebra known as group theory, the Harada–Norton groupHN 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 McLaughlin group McL is a sporadic simple group of order
In mathematical finite group theory, the Dempwolff group is a finite group of order 319979520 = 215·32·5·7·31, that is the unique nonsplit extension of by its natural module of order . The uniqueness of such a nonsplit extension was shown by Dempwolff (1972), and the existence by Thompson (1976), who showed using some computer calculations of Smith (1976) that the Dempwolff group is contained in the compact Lie group as the subgroup fixing a certain lattice in the Lie algebra of , and is also contained in the Thompson sporadic group (the full automorphism group of this lattice) as a maximal subgroup.
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
