# 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) |

#### Comments

For more information (e.g. character tables and maximal subgroups) see [a1].

#### References

[a1] | J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, R.A. Wilson, "Atlas of finite groups" , Clarendon Press (1985) |

**How to Cite This Entry:**

Mathieu group.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Mathieu_group&oldid=51328