Mathieu group

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In group theory, a topic in abstract algebra, 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 are the first sporadic groups to be discovered.

Sometimes the notation M₈, M₉, M₁₀, M₂₀, and M₂₁ is used for related groups (which act on sets of 8, 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).