History
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 automorphism groups of Steiner systems.
After the Mathieu groups no new sporadic groups were found until 1965, when the group J1 was discovered.
Read more about this topic: Mathieu Group
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