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Mathieu group. 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.
M24 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.
History and properties. M11 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. M11 is a sharply 4-transitive permutation group on 11 ...
M12 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).
M23 is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873). It is a 4-fold transitive permutation group on 23 objects. The Schur multiplier and the outer automorphism group are both trivial. Milgram (2000) calculated the integral cohomology, and showed in particular that M 23 has the unusual property that the first 4 ...
M22 is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873). It is a 3-fold transitive permutation group on 22 objects. The Schur multiplier of M 22 is cyclic of order 12, and the outer automorphism group has order 2. There are several incorrect statements about the 2-part of the Schur multiplier in the mathematical literature.
The Mathieu group M 12 is the automorphism group of a S(5,6,12) Steiner system; The Mathieu group M 22 is the unique index 2 subgroup of the automorphism group of a S(3,6,22) Steiner system; The Mathieu group M 23 is the automorphism group of a S(4,7,23) Steiner system; The Mathieu group M 24 is the automorphism group of a S(5,8,24) Steiner system.
The following table is a complete list of the 18 families of finite simple groups and the 26 sporadic simple groups, along with their orders. Any non-simple members of each family are listed, as well as any members duplicated within a family or between families.