Mathieu point stabilizers

Statement

In the natural permutation actions, a point stabilizer in $M_{12}$ is isomorphic to $M_{11}$ , one in $M_{23}$ is isomorphic to $M_{22}$ , and one in $M_{24}$ is isomorphic to $M_{23}$ . Hence there are injective group homomorphisms

\begin{gathered} M_{11}\hookrightarrow M_{12},\quad [M_{12}:M_{11}]=12,\\ M_{22}\hookrightarrow M_{23},\quad [M_{23}:M_{22}]=23,\\ M_{23}\hookrightarrow M_{24},\quad [M_{24}:M_{23}]=24. \end{gathered}

Each inclusion uses a chosen point and an isomorphism from the named source group to its stabilizer.

Why the indices are the degrees

The actions are transitive, so the orbit–stabilizer theorem identifies each set of left cosets with the relevant point set. Each action is at least doubly transitive, hence primitive, and its point stabilizer is maximal. Composing the last two embeddings also gives a copy of $M_{22}$ in $M_{24}$ of index $23\cdot24=552$ .

Scope

These are specified subgroup constructions. They do not classify all embeddings or all homomorphisms between the Mathieu groups, and the $M_{11}\subset M_{12}$ relation belongs to a separate stabilizer chain.

References

ATLAS M12G1-p12aB0, Properties: transitivity and point stabilizer.
ATLAS M23G1-p23B0, Properties: transitivity and point stabilizer.
ATLAS M24G1-p24B0, Properties: transitivity and point stabilizer.