Difference between revisions of "Mathieu group"
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A [[Finite group|finite group]] isomorphic to one of the five groups discovered by E. Mathieu . The series of Mathieu groups consists of the groups denoted by | A [[Finite group|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|Permutation group]]) on sets with 11, 12, 22, 23, and 24 elements, respectively. The groups | + | They are representable as permutation groups (cf. [[Permutation group|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|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|Steiner system]] | + | When considering a Mathieu group, one often uses (see ) its representation as the group of automorphisms of the corresponding [[Steiner system|Steiner system]] $ S( l, m, n) $, |
+ | i.e. of the set of $ n $ | ||
+ | elements in which there is distinguished a system of | ||
− | + | $$ | |
+ | {\left ( \begin{array}{c} | ||
+ | m \\ | ||
+ | l | ||
+ | \end{array} | ||
+ | \right ) ^ {-} 1 } {\left ( \begin{array}{c} | ||
+ | n \\ | ||
+ | l | ||
+ | \end{array} | ||
− | subsets, called blocks, consisting of | + | \right ) } |
+ | $$ | ||
+ | |||
+ | 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|Sporadic simple group]]). | The Mathieu groups were the first (and for over 80 years the only) known sporadic finite simple groups (cf. also [[Sporadic simple group|Sporadic simple group]]). | ||
Line 17: | Line 66: | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1a]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top"> E. Mathieu, "Sur la fonction cinq fois transitive des 24 quantités" ''J. Math. Pures Appl.'' , '''18''' (1873) pp. 25–46</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top"> E. Witt, "Die <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062770/m06277027.png" />-fach transitiven Gruppen von Matthieu" ''Abh. Math. Sem. Univ. Hamburg'' , '''12''' (1938) pp. 256–264</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top"> E. Witt, "Ueber Steinersche Systeme" ''Abh. Math. Sem. Univ. Hamburg'' , '''12''' (1938) pp. 265–275</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> V.D. Mazurov, "Finite groups" ''Itogi Nauk. i Tekhn. Algebra. Topol. Geom.'' , '''14''' (1976) pp. 5–56 (In Russian)</TD></TR></table> | <table><TR><TD valign="top">[1a]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top"> E. Mathieu, "Sur la fonction cinq fois transitive des 24 quantités" ''J. Math. Pures Appl.'' , '''18''' (1873) pp. 25–46</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top"> E. Witt, "Die <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062770/m06277027.png" />-fach transitiven Gruppen von Matthieu" ''Abh. Math. Sem. Univ. Hamburg'' , '''12''' (1938) pp. 256–264</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top"> E. Witt, "Ueber Steinersche Systeme" ''Abh. Math. Sem. Univ. Hamburg'' , '''12''' (1938) pp. 265–275</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> V.D. Mazurov, "Finite groups" ''Itogi Nauk. i Tekhn. Algebra. Topol. Geom.'' , '''14''' (1976) pp. 5–56 (In Russian)</TD></TR></table> | ||
− | |||
− | |||
====Comments==== | ====Comments==== |
Revision as of 07:59, 6 June 2020
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 ![]() |
[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=18116