| Algebraic structure → Group theory Group theory |
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In the area of modern algebra known as group theory, the Mathieu groupM11 is a sporadic simple group of order
M11 is one of the 26 sporadic groups and was introduced by Mathieu ( 1861 , 1873 ). It is the smallest sporadic group and, along with the other four Mathieu groups, the first to be discovered. The Schur multiplier and the outer automorphism group are both trivial.
M11 is a sharply 4-transitive permutation group on 11 objects. It admits many generating sets of permutations, such as the pair (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) of permutations used by the GAP computer algebra system.
M11 has a sharply 4-transitive permutation representation on 11 points. The point stabilizer is sometimes denoted by M10, and is a non-split extension of the form A6.2 (an extension of the group of order 2 by the alternating group A6). This action is the automorphism group of a Steiner system S(4,5,11). The induced action on unordered pairs of points gives a rank 3 action on 55 points.
M11 has a 3-transitive permutation representation on 12 points with point stabilizer PSL2(11). The permutation representations on 11 and 12 points can both be seen inside the Mathieu group M12 as two different embeddings of M11 in M12, exchanged by an outer automorphism.
The permutation representation on 11 points gives a complex irreducible representation in 10 dimensions. This is the smallest possible dimension of a faithful complex representation, though there are also two other such representations in 10 dimensions forming a complex conjugate pair.
M11 has two 5-dimensional irreducible representations over the field with 3 elements, related to the restrictions of 6-dimensional representations of the double cover of M12. These have the smallest dimension of any faithful linear representations of M11 over any field.
There are 5 conjugacy classes of maximal subgroups of M11 as follows:
The maximum order of any element in M11 is 11. Cycle structures are shown for the representations both of degree 11 and 12.
| Order | No. elements | Degree 11 | Degree 12 | |
|---|---|---|---|---|
| 1 = 1 | 1 = 1 | 111· | 112· | |
| 2 = 2 | 165 = 3 · 5 · 11 | 13·24 | 14·24 | |
| 3 = 3 | 440 = 23 · 5 · 11 | 12·33 | 13·33 | |
| 4 = 22 | 990 = 2 · 32 · 5 · 11 | 13·42 | 22·42 | |
| 5 = 5 | 1584 = 24 · 32 · 11 | 1·52 | 12·52 | |
| 6 = 2 · 3 | 1320 = 23 · 3 · 5 · 11 | 2·3·6 | 1·2·3·6 | |
| 8 = 23 | 990 = 2 · 32 · 5 · 11 | 1·2·8 | 4·8 | power equivalent |
| 990 = 2 · 32 · 5 · 11 | 1·2·8 | 4·8 | ||
| 11 = 11 | 720 = 24 · 32 · 5 | 11 | 1·11 | power equivalent |
| 720 = 24 · 32 · 5 | 11 | 1·11 |
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 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 the area of modern algebra known as group theory, the Lyons groupLy or Lyons-Sims groupLyS is a sporadic simple group of order
In mathematics, a Niemeier lattice is one of the 24 positive definite even unimodular lattices of rank 24, which were classified by Hans-Volker Niemeier. Venkov (1978) gave a simplified proof of the classification. Witt (1941) mentions that he found more than 10 such lattices, but gives no further details. One example of a Niemeier lattice is the Leech lattice found in 1967.
In mathematics, a Gelfand pair is a pair (G,K ) consisting of a group G and a subgroup K (called an Euler subgroup of G) that satisfies a certain property on restricted representations. The theory of Gelfand pairs is closely related to the topic of spherical functions in the classical theory of special functions, and to the theory of Riemannian symmetric spaces in differential geometry. Broadly speaking, the theory exists to abstract from these theories their content in terms of harmonic analysis and representation theory.
In the area of modern algebra known as group theory, the Janko groupJ1 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 groupM23 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 the area of modern algebra known as group theory, the Fischer groupFi22 is a sporadic simple group of order
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
