# Mathieu group

A finite group isomorphic to one of the five groups discovered by E. Mathieu . The series of Mathieu groups consists of the groups denoted by

They are representable as permutation groups (cf. Permutation group) on sets with 11, 12, 22, 23, and 24 elements, respectively. The groups and are five-fold transitive. is realized naturally as the stabilizer in of an element of the set on which acts; similarly, and are stabilizers of elements of and , respectively. The Mathieu groups have the respective orders

When considering a Mathieu group, one often uses (see ) its representation as the group of automorphisms of the corresponding Steiner system , i.e. of the set of elements in which there is distinguished a system of

subsets, called blocks, consisting of elements of the set, and such that every set of elements is contained in one and only one block. An automorphism of a Steiner system is defined as a permutation of the set of its elements which takes blocks into blocks. The list of Mathieu groups and corresponding Steiner systems for which they are automorphism groups is as follows: ; ; ; ; .

The Mathieu groups were the first (and for over 80 years the only) known sporadic finite simple groups (cf. also Sporadic simple group).

#### References

 [1a] E. Mathieu, "Mémoire sur l'étude des fonctions de plusieures quantités, sur la manière de les formes et sur les substitutions qui les laissant invariables" J. Math. Pures Appl. , 6 (1861) pp. 241–323 [1b] E. Mathieu, "Sur la fonction cinq fois transitive des 24 quantités" J. Math. Pures Appl. , 18 (1873) pp. 25–46 [2a] E. Witt, "Die -fach transitiven Gruppen von Matthieu" Abh. Math. Sem. Univ. Hamburg , 12 (1938) pp. 256–264 [2b] E. Witt, "Ueber Steinersche Systeme" Abh. Math. Sem. Univ. Hamburg , 12 (1938) pp. 265–275 [3] V.D. Mazurov, "Finite groups" Itogi Nauk. i Tekhn. Algebra. Topol. Geom. , 14 (1976) pp. 5–56 (In Russian)