| Algebraic structure → Group theory Group theory |
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In the area of modern algebra known as group theory, the Mathieu groupM23 is a sporadic simple group of order
M23 is one of the 26 sporadic groups and was introduced by Mathieu ( 1861 , 1873 ). It is a 4-fold transitive permutation group on 23 objects. The Schur multiplier and the outer automorphism group are both trivial.
Milgram (2000) calculated the integral cohomology, and showed in particular that M23 has the unusual property that the first 4 integral homology groups all vanish.
The inverse Galois problem seems to be unsolved for M23. In other words, no polynomial in Z[x] seems to be known to have M23 as its Galois group. The inverse Galois problem is solved for all other sporadic simple groups.
Let F211 be the finite field with 211 elements. Its group of units has order 211 − 1 = 2047 = 23 · 89, so it has a cyclic subgroup C of order 23.
The Mathieu group M23 can be identified with the group of F2-linear automorphisms of F211 that stabilize C. More precisely, the action of this automorphism group on C can be identified with the 4-fold transitive action of M23 on 23 objects.
M23 is the point stabilizer of the action of the Mathieu group M24 on 24 points, giving it a 4-transitive permutation representation on 23 points with point stabilizer the Mathieu group M22.
M23 has 2 different rank 3 actions on 253 points. One is the action on unordered pairs with orbit sizes 1+42+210 and point stabilizer M21.2, and the other is the action on heptads with orbit sizes 1+112+140 and point stabilizer 24.A7.
The integral representation corresponding to the permutation action on 23 points decomposes into the trivial representation and a 22-dimensional representation. The 22-dimensional representation is irreducible over any field of characteristic not 2 or 23.
Over the field of order 2, it has two 11-dimensional representations, the restrictions of the corresponding representations of the Mathieu group M24.
There are 7 conjugacy classes of maximal subgroups of M23 as follows:
| Order | No. elements | Cycle structure | |
|---|---|---|---|
| 1 = 1 | 1 | 123 | |
| 2 = 2 | 3795 = 3 · 5 · 11 · 23 | 1728 | |
| 3 = 3 | 56672 = 25 · 7 · 11 · 23 | 1536 | |
| 4 = 22 | 318780 = 22 · 32 · 5 · 7 · 11 · 23 | 132244 | |
| 5 = 5 | 680064 = 27 · 3 · 7 · 11 · 23 | 1354 | |
| 6 = 2 · 3 | 850080 = 25 · 3 · 5 · 7 · 11 · 23 | 1·223262 | |
| 7 = 7 | 728640 = 26 · 32 · 5 · 11 · 23 | 1273 | power equivalent |
| 728640 = 26 · 32 · 5 · 11 · 23 | 1273 | ||
| 8 = 23 | 1275120 = 24 · 32 · 5 · 7 · 11 · 23 | 1·2·4·82 | |
| 11 = 11 | 927360= 27 · 32 · 5 · 7 · 23 | 1·112 | power equivalent |
| 927360= 27 · 32 · 5 · 7 · 23 | 1·112 | ||
| 14 = 2 · 7 | 728640= 26 · 32 · 5 · 11 · 23 | 2·7·14 | power equivalent |
| 728640= 26 · 32 · 5 · 11 · 23 | 2·7·14 | ||
| 15 = 3 · 5 | 680064= 27 · 3 · 7 · 11 · 23 | 3·5·15 | power equivalent |
| 680064= 27 · 3 · 7 · 11 · 23 | 3·5·15 | ||
| 23 = 23 | 443520= 27 · 32 · 5 · 7 · 11 | 23 | power equivalent |
| 443520= 27 · 32 · 5 · 7 · 11 | 23 |
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group acts on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. In particular, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron.
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group defined over a finite set of symbols consists of the permutations that can be performed on the symbols. Since there are such permutation operations, the order of the symmetric group is .
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
246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71
= 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000
≈ 8×1053.
In group theory, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple 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 were 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 Mathieu groupM11 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 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 mathematical finite group theory, a rank 3 permutation group acts transitively on a set such that the stabilizer of a point has 3 orbits. The study of these groups was started by Higman. Several of the sporadic simple groups were discovered as rank 3 permutation groups.
In the area of modern algebra known as group theory, the Fischer groupFi24 or F24′ 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
