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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.
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.
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.
The diagram shows the subquotient relations between the 26 sporadic groups. A connecting line means the lower group is subquotient of the upper – and no sporadic subquotient in between. The generations of Robert Griess: 1st, 2nd, 3rd, Pariah. Mathieu groups M 11, M 12, M 22, M 23, M 24; Janko groups J 1, J 2 or HJ, J 3 or HJM, J 4; Conway ...
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).
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra.
Its group of units has order 2 11 − 1 = 2047 = 23 · 89, so it has a cyclic subgroup C of order 23. The Mathieu group M 23 can be identified with the group of F 2-linear automorphisms of F 2 11 that stabilize C. More precisely, the action of this automorphism group on C can be identified with the 4-fold transitive action of M 23 on 23 objects.