In the area of modern algebra known as
group theory, the Mathieu group ''M
11'' 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
43
2511 = 111098 = 7920.
History and properties
''M
11'' is one of the 26 sporadic groups and was introduced by . 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.
''M
11'' is a
sharply 4-transitive permutation group
In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to it ...
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.
Representations
M
11 has a sharply 4-transitive permutation representation on 11 points. The point stabilizer is sometimes denoted by M
10, and is a non-split extension of the form A
6.2 (an extension of the group of order 2 by the alternating group A
6). 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.
M
11 has a 3-transitive permutation representation on 12 points with point stabilizer PSL
2(11). The permutation representations on 11 and 12 points can both be seen inside the
Mathieu group M12
In the area of modern algebra known as group theory, the Mathieu group ''M12'' is a sporadic simple group of order
: 12111098 = 2633511 = 95040.
History and properties
''M12'' is one of the 26 sporadic groups and was introduc ...
as two different embeddings of M
11 in M
12, 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.
M
11 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 M
12. These have the smallest dimension of any faithful linear representations of M
11 over any field.
Maximal subgroups
There are 5 conjugacy classes of maximal subgroups of ''M
11'' as follows:
* M
10, order 720, one-point stabilizer in representation of degree 11
* PSL(2,11), order 660, one-point stabilizer in representation of degree 12
* M
9:2, order 144, stabilizer of a 9 and 2 partition.
* S
5, order 120, orbits of 5 and 6
: Stabilizer of block in the S(4,5,11) Steiner system
*
Q:S
3, order 48, orbits of 8 and 3
: Centralizer of a quadruple transposition
: Isomorphic to GL(2,3).
Conjugacy classes
The maximum order of any element in M
11 is 11. Cycle structures are shown for the representations both of degree 11 and 12.
References
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External links
MathWorld: Mathieu Groups Atlas of Finite Group Representations: M11
{{DEFAULTSORT:Mathieu Group M11
Sporadic groups