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The only 4-transitive groups are the symmetric groups S k for k at least 4, the alternating groups A k for k at least 6, and the Mathieu groups M 24, M 23, M 12, and M 11. ( Cameron 1999 , p. 110) The full proof requires the classification of finite simple groups , but some special cases have been known for much longer.
M 11 is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873). It is the smallest sporadic group and, along with the other four Mathieu groups, the first to be discovered. The Schur multiplier and the outer automorphism group are both trivial. M 11 is a sharply 4-transitive permutation group on 11 objects.
M 24 is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873). It is a 5-transitive permutation group on 24 objects. The Schur multiplier and the outer automorphism group are both trivial. The Mathieu groups can be constructed in various ways. Initially, Mathieu and others constructed them as permutation groups.
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M 12 is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873).It is a sharply 5-transitive permutation group on 12 objects. Burgoyne & Fong (1968) showed that the Schur multiplier of M 12 has order 2 (correcting a mistake in (Burgoyne & Fong 1966) where they incorrectly claimed it has order 1).
Since it opened in February 1968, the current incarnation of MSG has hosted the Democratic National Convention three times — in 1976, 1980 and 1992 — and the Republican National Convention ...
This action fixes a Steiner system S(3,6,22) with 77 hexads, whose full automorphism group is the automorphism group M 22.2 of M 22. M 22 has three rank 3 permutation representations : one on the 77 hexads with point stabilizer 2 4 :A 6 , and two rank 3 actions on 176 heptads that are conjugate under an outer automorphism and have point ...
In mathematics, the Mathieu groupoid M 13 is a groupoid acting on 13 points such that the stabilizer of each point is the Mathieu group M 12. It was introduced by Conway ( 1987 , 1997 ) and studied in detail by Conway, Elkies & Martin (2006) .