Mathieu groups
In the area of modern algebra known as group theory, the Mathieu groups are the five sporadic simple groups M11, M12, M22, M23 and M24 introduced by Mathieu (1861, 1873). They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objects. They were the first sporadic groups to be discovered.
Sometimes the notation M9, M10, M20 and M21 is used for related groups (which act on sets of 9, 10, 20, and 21 points, respectively), namely the stabilizers of points in the larger groups. While these are not sporadic simple groups, they are subgroups of the larger groups and can be used to construct the larger ones. John Conway has shown that one can also extend this sequence up, obtaining the Mathieu groupoid M13 acting on 13 points. M21 is simple, but is not a sporadic group, being isomorphic to PSL(3,4).
Mathieu (1861, p.271) introduced the group M12 as part of an investigation of multiply transitive permutation groups, and briefly mentioned (on page 274) the group M24, giving its order. In Mathieu (1873) he gave further details, including explicit generating sets for his groups, but it was not easy to see from his arguments that the groups generated are not just alternating groups, and for several years the existence of his groups was controversial. Miller (1898) even published a paper mistakenly claiming to prove that M24 does not exist, though shortly afterwards in (Miller 1900) he pointed out that his proof was wrong, and gave a proof that the Mathieu groups are simple. Witt (1938a, 1938b) finally removed the doubts about the existence of these groups, by constructing them as successive transitive extensions of permutation groups, as well as automorphism groups of Steiner systems.
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