Mathieu group M11 - Knowlpedia

The Mathieu group $M_{11}$ is the subgroup generated by the following permutations in the symmetric group $S_{11}$ :

\begin{aligned}a&=(2\,10)(4\,11)(5\,7)(8\,9),\\ b&=(1\,4\,3\,8)(2\,5\,6\,9). \end{aligned}

Thus $M_{11}=\langle a,b\rangle$ , 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_{11}|=7920=2^{4}\cdot3^{2}\cdot5\cdot11.

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 $11$ points is $4$ -transitive: any ordered $4$ -tuple of distinct points can be carried to any other such tuple. The element doing so is unique; this is called sharp transitivity.

The orbit of $\{1,2,3,4,5\}$ is a collection of $66$ blocks of size $5$ . Every $4$ -element subset lies in exactly one block, so these blocks form a Steiner system $S(4,5,11)$ . The generators preserve the system.

The natural point stabilizers connect the Mathieu groups.

References

ATLAS, Mathieu group M11, order heading.
ATLAS, permutation model M11G1-p11B0, “Properties,” preserved Steiner system and “Download”; exact generators a and b.