Mathieu group M23 - Knowlpedia

The Mathieu group $M_{23}$ is the subgroup generated by the following permutations in the symmetric group $S_{23}$ :

\begin{aligned}a&=(1\,2)(3\,4)(7\,8)(9\,10)\\ &\quad{}(13\,14)(15\,16)(19\,20)(21\,22),\\ b&=(1\,16\,11\,3)(2\,9\,21\,12)(4\,5\,8\,23)(6\,22\,14\,18)\\ &\quad{}(13\,20)(15\,17). \end{aligned}

Thus $M_{23}=\langle a,b\rangle$ , with permutation composition as the group operation. Cycles list successive images, unlisted points are fixed, and products act from right to left.

Order and structure

This is a nonabelian finite simple group, with

|M_{23}|=10200960=2^{7}\cdot3^{2}\cdot5\cdot7\cdot11\cdot23.

It belongs to the Mathieu display block of the sporadic groups. This grouping records the ATLAS organization, without asserting a direct embedding into another displayed group.

The natural permutation action

The action on $23$ points is $4$ -transitive: any ordered $4$ -tuple of distinct points can be carried to any other such tuple.

The orbit of $\{1,2,3,4,5,11,17\}$ is a collection of $253$ blocks of size $7$ . Every $4$ -element subset lies in exactly one block, so these blocks form a Steiner system $S(4,7,23)$ . The generators preserve the system.

The natural point stabilizers connect the Mathieu groups.

References

ATLAS, Mathieu group M23, order heading.
ATLAS, permutation model M23G1-p23B0, “Properties,” preserved Steiner system and “Download”; exact generators a and b.