# 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

$$M _ {11} , M _ {12} , M _ {22} , M _ {23} , M _ {24} .$$

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

$$7 920 , 95 040 , 443 520 , 10 200 960 , 244 823 040.$$

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

$${\left ( \begin{array}{c} m \\ l \end{array} \right ) ^ {-1} } {\left ( \begin{array}{c} n \\ l \end{array} \right ) }$$

subsets, called blocks, consisting of $m$ elements of the set, and such that every set of $l$ 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: $M _ {11}$— $S( 4, 5, 11)$; $M _ {12}$— $S( 5, 6, 12)$; $M _ {22}$— $S( 3, 6, 22)$; $M _ {23}$— $S( 4, 7, 23)$; $M _ {24}$— $S( 5, 8, 24)$.

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 $5$-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)