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''
23 is a
sporadic simple group 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
73
2571123 = 10200960
: ≈ 1 × 10
7.
History and properties
''M''
23 is one of the 26 sporadic groups and was introduced by . It is a 4-fold transitive
permutation group on 23 objects. The
Schur multiplier and the
outer automorphism group are both
trivial.
calculated the integral cohomology, and showed in particular that M
23 has the unusual property that the first 4 integral homology groups all vanish.
The
inverse Galois problem seems to be unsolved for M
23. In other words, no
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
in Z
'x''seems to be known to have M
23 as its
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
. The inverse Galois problem is solved for all other sporadic simple groups.
Construction using finite fields
Let be the
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...
with 2
11 elements. Its
group of units has order − 1 = 2047 = 23 · 89, so it has a
cyclic subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...
of order 23.
The Mathieu group M
23 can be identified with the group of -
linear automorphisms of that stabilize . More precisely, the
action of this
automorphism group on can be identified with the 4-fold transitive action of M
23 on 23 objects.
Representations
M
23 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.
M
23 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 M
21.2, and the other is the action on heptads with orbit sizes 1+112+140 and point stabilizer 2
4.A
7.
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.
Maximal subgroups
There are 7 conjugacy classes of maximal subgroups of ''M''
23 as follows:
*
M22, order 443520
* PSL(3,4):2, order 40320, orbits of 21 and 2
* 2
4:A
7, order 40320, orbits of 7 and 16
: Stabilizer of W
23 block
* A
8, order 20160, orbits of 8 and 15
*
M11, order 7920, orbits of 11 and 12
* (2
4:A
5):S
3 or M
20:S
3, order 5760, orbits of 3 and 20 (5 blocks of 4)
: One-point stabilizer of the sextet group
* 23:11, order 253, simply transitive
Conjugacy classes
References
*
*
* Reprinted in
*
*
*
*
*
*
*
*
*
*
*
External links
MathWorld: Mathieu GroupsAtlas of Finite Group Representations: M23
{{DEFAULTSORT:Mathieu Group M23
Sporadic groups