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 ''M
12'' is a
sporadic simple group
In the mathematical classification of finite simple groups, there are a number of 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.
A simpl ...
of
order
Order, ORDER or Orders may refer to:
* A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
...
: 95,040 = 12111098 = 2
63
3511.
History and properties
''M
12'' is one of the 26 sporadic groups and was introduced by . It is a
sharply 5-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 ...
on 12 objects. showed that 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 \ope ...
of M
12 has order 2 (correcting a mistake in where they incorrectly claimed it has order 1).
The double cover had been implicitly found earlier by , who showed that M
12 is a subgroup of the
projective linear group
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associa ...
of dimension 6 over the
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
with 3 elements.
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 ...
has order 2, and the full automorphism group M
12.2 is contained in M
24 as the stabilizer of a pair of complementary dodecads of 24 points, with outer automorphisms of M
12 swapping the two dodecads.
Representations
calculated the complex character table of M
12.
M
12 has a strictly 5-transitive permutation representation on 12 points, whose point stabilizer is the
Mathieu group M11. Identifying the 12 points with the projective line over the field of 11 elements, M
12 is generated by the permutations of PSL
2(11) together with the permutation (2,10)(3,4)(5,9)(6,7). This permutation representation preserves 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(5,6,12) of 132 special hexads, such that each pentad is contained in exactly 1 special hexad, and the hexads are the supports of the weight 6 codewords of the extended
ternary Golay code
In coding theory, the ternary Golay codes are two closely related error-correcting codes.
The code generally known simply as the ternary Golay code is an 1, 6, 53-code, that is, it is a linear code over a ternary alphabet; the relative distan ...
. In fact M
12 has two inequivalent actions on 12 points, exchanged by an outer automorphism; these are analogous to the two inequivalent actions of the symmetric group ''S''
6 on 6 points.
The double cover 2.M
12 is the automorphism group of the extended
ternary Golay code
In coding theory, the ternary Golay codes are two closely related error-correcting codes.
The code generally known simply as the ternary Golay code is an 1, 6, 53-code, that is, it is a linear code over a ternary alphabet; the relative distan ...
, a dimension 6 length 12 code over the field of order 3 of minimum weight 6. In particular the double cover has an irreducible 6-dimensional representation over the field of 3 elements.
The double cover 2.M
12 is the automorphism group of any 12×12
Hadamard matrix
In mathematics, an Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometry, geometric terms, this means that each pair of r ...
.
M
12 centralizes an element of order 11 in the
monster group
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; it has order
:
: = 2463205976112133171923293 ...
, as a result of which it acts naturally on a
vertex algebra
In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications, vertex operator algebras have proven usef ...
over the field with 11 elements, given as the
Tate cohomology of the
monster vertex algebra.
Maximal subgroups
There are 11 conjugacy classes of maximal subgroups of M
12, 6 occurring in automorphic pairs, as follows:
Conjugacy classes
The cycle shape of an element and its conjugate under an outer automorphism are related in the following way: the union of the two cycle shapes is balanced, in other words invariant under changing each ''n''-cycle to an ''N''/''n'' cycle for some integer ''N''.
In art
Composer
Olivier Messiaen
Olivier Eugène Prosper Charles Messiaen (, ; ; 10 December 1908 – 27 April 1992) was a French composer, organist, and ornithology, ornithologist. One of the major composers of the 20th-century classical music, 20th century, he was also an ou ...
used permutations in parts of his 1949-1950 ''
Quatre Études de rythme''. Some of these permutations correspond to ''M''
12 predating the discovery by Mathieu.
References
*
*
*
*
*
* Reprinted in
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
External links
MathWorld: Mathieu GroupsAtlas of Finite Group Representations: M12
{{DEFAULTSORT:Mathieu Group M12
Sporadic groups