Ciro Santilli @cirosantilli 40

Contains the first sporadic groups discovered by far: 11 and 12 in 1861, and 22, 23 and 24 in 1973. And therefore presumably the simplest! The next sporadic ones discovered were the Janko groups, only in 1965!

Each $M_n$ is a permutation group on $n$ elements. There isn't an obvious algorithmic relationship between $n$ and the actual group.

TODO initial motivation? Why did Mathieu care about k-transitive groups?

Their; k-transitive group properties seem to be the main characterization, according to Wikipedia:Looking at the classification of k-transitive groups we see that the Mathieu groups are the only families of 4 and 5 transitive groups other than symmetric groups and alternating groups. 3-transitive is not as nice, so let's just say it is the stabilizer of $M_{23}$ and be done with it.

Suppose we have a given permutation group that acts on a set of n elements.

If we pick k elements of the set, the stabilizer subgroup of those k elements is a subgroup of the given permutation group that keeps those elements unchanged.

Note that an analogous definition can be given for non-finite groups. Also note that the case for all finite groups is covered by the permutation definition since all groups are isomorphic to a subgroup of the symmetric group

TODO existence and uniqueness. Existence is obvious for the identity permutation, but proper subgroup likely does not exist in general.

Bibliography: