Editing Mathieu group (section)

== History ==

{{harvtxt|Mathieu|1861|loc=p.271}} introduced the group ''M''<sub>12</sub> as part of an investigation of multiply transitive permutation groups, and briefly mentioned (on page 274) the group ''M''<sub>24</sub>, giving its order. In {{harvtxt|Mathieu|1873}} he gave further details, including explicit [[Generating set of a group|generating sets]] for his groups, but it was not easy to see from his arguments that the groups generated are not just [[alternating group]]s, and for several years the existence of his groups was controversial.  {{harvtxt|Miller|1898}} even published a paper mistakenly claiming to prove that ''M''<sub>24</sub> does not exist, though shortly afterwards in {{harv|Miller|1900}} he pointed out that his proof was wrong, and gave a proof that the Mathieu groups are simple. {{Harvs|txt|authorlink=Robert Daniel Carmichael|last=Carmichael|year=1931}} and later  {{harvs|txt|last=Witt|year1=1938a|year2=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 system]]s.

After the Mathieu groups, no new sporadic groups were found until 1965, when the group [[Janko group J1|J<sub>1</sub>]] was discovered.