In the area of modern algebra known as group theory, the Mathieu group M<sub>23</sub> is a sporadic simple group of order
M<sub>23</sub> 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<sub>23</sub> has the unusual property that the first 4 integral homology groups all vanish.
The inverse Galois problem seems to be unsolved for M<sub>23</sub>. In other words, no polynomial in Z[x] seems to be known to have M<sub>23</sub> as its Galois group. The inverse Galois problem is solved for all other sporadic simple groups.
Let be the finite field with 2<sup>11</sup> elements. Its group of units has order â 1 = 2047 = 23 ÷ 89, so it has a cyclic subgroup of order 23.
The Mathieu group M<sub>23</sub> 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<sub>23</sub> on 23 objects.
M<sub>23</sub> 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<sub>23</sub> 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<sub>21</sub>.2, and the other is the action on heptads with orbit sizes 1+112+140 and point stabilizer 2<sup>4</sup>.A<sub>7</sub>.
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 M<sub>23</sub> as follows: