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
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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
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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
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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 ![]() |
[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) |
Mathieu group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mathieu_group&oldid=47792