Mathieu Group M11
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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 Mathieu group ''M11'' is a sporadic simple group of order :   2432511 = 111098 = 7920.


History and properties

''M11'' 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 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 \oper ...
and 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 t ...
are both
trivial Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense. Latin Etymology The ancient Romans used the word ''triviae'' to describe where one road split or forked ...
. ''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 GAP (Groups, Algorithms and Programming) is a computer algebra system for computational discrete algebra with particular emphasis on computational group theory. History GAP was developed at Lehrstuhl D für Mathematik (LDFM), Rheinisch-Westfà ...
.


Representations

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 250px, thumbnail, The Fano plane is a Steiner triple system S(2,3,7). The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line. In combinatorial mathematics, a Steiner system (named after Jakob Steiner) ...
S(4,5,11). The induced action on unordered pairs of points gives a
rank 3 action Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...
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.


Maximal subgroups

There are 5 conjugacy classes of maximal subgroups of ''M11'' as follows: * M10, order 720, one-point stabilizer in representation of degree 11 * PSL(2,11), order 660, one-point stabilizer in representation of degree 12 * M9:2, order 144, stabilizer of a 9 and 2 partition. * S5, order 120, orbits of 5 and 6 : Stabilizer of block in the S(4,5,11) Steiner system * Q:S3, 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 M11 is 11. Cycle structures are shown for the representations both of degree 11 and 12.


References

* * * Reprinted in * * * * * * * * * * * * *


External links


MathWorld: Mathieu Groups

Atlas of Finite Group Representations: M11
{{DEFAULTSORT:Mathieu Group M11 Sporadic groups