The Mathieu groups and designs

In the early 80-ties it became clear that the classification of the non-abelian finite simple groups was complete. Among the finite simple groups we find several families: the alternating groups and various families of groups of Lietype and their twisted analogues. Besides these families there exist 26 sporadic finite simple groups. In these notes we consider five sporadic simple groups, the Mathieu groups Mi, i ∈ {11, 12, 22, 23, 24}, as well as some of their geometries. The Mathieu groups were discovered by the French mathematician Émile Mathieu (1835–1890), who also discovered the large Mathieu groups M22, M23 and M24. See [12, 13, 14]. They are remarkable groups: for example, apart from the symmetric and alternating groups, M12 and M24 are the only 5-transitive permutation groups. After Mathieu’s discovery of these five sporadic simple groups it took almost a century before the sixth sporadic simple group was found. Not only are the five Mathieu groups among the 26 sporadic simple groups, they are also closely related to almost all the other sporadics. The geometries we will describe in these notes are also closely connected to various geometries associated to other sporadic simple groups.