Difference between revisions of "Mathieu group"
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Sometimes the notation | In [[mathematics]], the '''Mathieu groups''' [[Mathieu_group_M11|M<sub>11</sub>]], [[Mathieu_group_M12|M<sub>12</sub>]], [[Mathieu_group_M22|M<sub>22</sub>]], [[Mathieu_group_M23|M<sub>23</sub>]], [[Mathieu_group_M24|M<sub>24</sub>]], introduced by {{harvs|txt |authorlink=Émile Léonard Mathieu |last=Mathieu |year1=1861 |year2=1873}}, are multiply transitive [[permutation group]]s on 11, 12, 22, 23 or 24 objects. They were the first [[sporadic simple groups]] discovered. | ||
Sometimes the notation M<sub>10</sub>, M<sub>20</sub> and M<sub>21</sub> is used for related groups (which act on sets of 10, 20, and 21 points, respectively), namely the stabilizers of points in the larger groups. While these are not sporadic simple groups, they are subgroups of the larger groups and can be used to construct the larger ones. [[John Horton Conway|John Conway]] has shown that one can also extend this sequence up, obtaining the [[Mathieu groupoid |Mathieu groupoid M<sub>13</sub>]] acting on 13 points. M<sub>21</sub> is simple, but is not a sporadic group, being isomorphic to PSL(3,4). | |||
== History == | == History == | ||
{{harvtxt|Mathieu|1861|loc=p.271}} introduced the group ''M''<sub>12</sub> as part of an investigation of multiply transitive permutation groups, and briefly mentioned (on page 274) the group ''M''<sub>24</sub>, giving its order. In {{harvtxt|Mathieu|1873}} he gave further details, including explicit generating sets for his groups, but it was not easy to see from his arguments that the groups generated are not just alternating groups, and for several years the existence of his groups was controversial. {{harvtxt|Miller|1898}} even published a paper mistakenly claiming to prove that ''M''<sub>24</sub> does not exist, though shortly afterwards in {{harv|Miller|1900}} he pointed out that his proof was wrong, and gave a proof that the Mathieu groups are simple. {{harvs|txt|last=Witt|year1=1938a|year2=1938b}} finally removed the doubts about the existence of these groups, by constructing them as automorphism groups of [[Steiner system]]s. | {{harvtxt|Mathieu|1861|loc=p.271}} introduced the group ''M''<sub>12</sub> as part of an investigation of multiply transitive permutation groups, and briefly mentioned (on page 274) the group ''M''<sub>24</sub>, giving its order. In {{harvtxt|Mathieu|1873}} he gave further details, including explicit [[Generating set of a group|generating sets]] for his groups, but it was not easy to see from his arguments that the groups generated are not just [[Alternating group|alternating groups]], and for several years the existence of his groups was controversial. {{harvtxt|Miller|1898}} even published a paper mistakenly claiming to prove that ''M''<sub>24</sub> does not exist, though shortly afterwards in {{harv|Miller|1900}} he pointed out that his proof was wrong, and gave a proof that the Mathieu groups are simple. {{harvs|txt|last=Witt|year1=1938a|year2=1938b}} finally removed the doubts about the existence of these groups, by constructing them as automorphism groups of [[Steiner system]]s. | ||
After the Mathieu groups no new sporadic groups were found until 1965, when the group [[Janko group J1|J<sub>1</sub>]] was discovered. | After the Mathieu groups no new sporadic groups were found until 1965, when the group [[Janko group J1|J<sub>1</sub>]] was discovered. | ||
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M<sub>24</sub> is 5-transitive, and M<sub>12</sub> is sharply 5-transitive, with the other Mathieu groups (simple or not) being the subgroups corresponding to stabilizers of ''m'' points, and accordingly of lower transitivity (M<sub>23</sub> is 4-transitive, etc.). | M<sub>24</sub> is 5-transitive, and M<sub>12</sub> is sharply 5-transitive, with the other Mathieu groups (simple or not) being the subgroups corresponding to stabilizers of ''m'' points, and accordingly of lower transitivity (M<sub>23</sub> is 4-transitive, etc.). | ||
The only 4-transitive groups are the [[symmetric group]]s S<sub>''k''</sub> for ''k'' at least 4, the [[alternating group]]s A<sub>''k''</sub> for ''k'' at least 6, and the Mathieu groups M<sub>24</sub>, M<sub>23</sub>, M<sub>12</sub> and M<sub>11</sub>. {{harv|Cameron|1999|loc= p. 110}} The full proof requires the [[classification of finite simple groups]], but some special cases have been known for much longer. | The only 4-transitive groups are the [[symmetric group]]s S<sub>''k''</sub> for ''k'' at least 4, the [[alternating group]]s A<sub>''k''</sub> for ''k'' at least 6, and the Mathieu groups [[Mathieu_group_M24|M<sub>24</sub>]], [[Mathieu_group_M23|M<sub>23</sub>]], [[Mathieu_group_M12|M<sub>12</sub>]] and [[Mathieu_group_M11|M<sub>11</sub>]]. {{harv|Cameron|1999|loc= p. 110}} The full proof requires the [[classification of finite simple groups]], but some special cases have been known for much longer. | ||
It is a classical result of [[Camille Jordan|Jordan]] that the [[symmetric group|symmetric]] and [[alternating group]]s (of degree ''k'' and ''k'' + 2 respectively), and M<sub>12</sub> and M<sub>11</sub> are the only ''sharply'' ''k''-transitive permutation groups for ''k'' at least 4. | It is a classical result of [[Camille Jordan|Jordan]] that the [[symmetric group|symmetric]] and [[alternating group]]s (of degree ''k'' and ''k'' + 2 respectively), and M<sub>12</sub> and M<sub>11</sub> are the only ''sharply'' ''k''-transitive permutation groups for ''k'' at least 4. | ||
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===Permutation groups=== | ===Permutation groups=== | ||
M<sub>12</sub> has a simple subgroup of order 660, a maximal subgroup. That subgroup can be represented as a linear fractional group on the [[field (mathematics)|field]] F<sub>11</sub> of 11 elements. With | M<sub>12</sub> has a simple subgroup of order 660, a maximal subgroup. That subgroup can be represented as a linear fractional group on the [[field (mathematics)|field]] F<sub>11</sub> of 11 elements. With −1 written as '''a''' and infinity as '''b''', two standard generators are (0123456789a) and (0b)(1a)(25)(37)(48)(69). A third generator giving M<sub>12</sub> sends an element ''x'' of ''F''<sub>11</sub> to 4''x''<sup>2</sup> − 3''x''<sup>7</sup>; as a permutation that is (26a7)(3945). The stabilizer of 4 points is a [[quaternion group]]. | ||
Likewise M<sub>24</sub> has a maximal simple subgroup of order 6072 and this can be represented as a linear fractional group on the field F<sub>23</sub>. One generator adds 1 to each element (leaving the point N at infinity fixed), i. e. (0123456789ABCDEFGHIJKLM)(N), and the other is the [[order reversing permutation]], (0N)(1M)(2B)(3F)(4H)(59)(6J)(7D)(8K)(AG)(CL)(EI). A third generator giving M<sub>24</sub> sends an element x of F<sub>23</sub> to | Likewise M<sub>24</sub> has a maximal simple subgroup of order 6072 and this can be represented as a linear fractional group on the field F<sub>23</sub>. One generator adds 1 to each element (leaving the point ''N'' at infinity fixed), i. e. (0123456789ABCDEFGHIJKLM)(''N''), and the other is the [[order reversing permutation]], (0N)(1M)(2B)(3F)(4H)(59)(6J)(7D)(8K)(AG)(CL)(EI). A third generator giving M<sub>24</sub> sends an element ''x'' of F<sub>23</sub> to 4''x''<sup>4</sup> − 3''x''<sup>15</sup> (which sends perfect squares via <math> x^4 </math> and non-perfect squares via <math> 7 x^4</math>); computation shows that as a permutation this is (2G968)(3CDI4)(7HABM)(EJLKF). | ||
These constructions were cited by {{harvtxt|Carmichael|1956|loc= pp.151, 164, 263}}. {{harvtxt|Dixon|Mortimer|1996|loc=p.209}} ascribe the permutations to Mathieu. | These constructions were cited by {{harvtxt|Carmichael|1956|loc= pp. 151, 164, 263}}. {{harvtxt|Dixon|Mortimer|1996|loc=p.209}} ascribe the permutations to Mathieu. | ||
=== Automorphism groups of Steiner systems === | === Automorphism groups of Steiner systems === | ||
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There exists [[up to]] [[Equivalence relation|equivalence]] a unique S(5,8,24) [[Steiner system]] '''W<sub>24</sub>''' (the [[Witt design]]). The group M<sub>24</sub> is the automorphism group of this Steiner system; that is, the set of permutations which map every block to some other block. The subgroups M<sub>23</sub> and M<sub>22</sub> are defined to be the stabilizers of a single point and two points respectively. | There exists [[up to]] [[Equivalence relation|equivalence]] a unique S(5,8,24) [[Steiner system]] '''W<sub>24</sub>''' (the [[Witt design]]). The group M<sub>24</sub> is the automorphism group of this Steiner system; that is, the set of permutations which map every block to some other block. The subgroups M<sub>23</sub> and M<sub>22</sub> are defined to be the stabilizers of a single point and two points respectively. | ||
Similarly, there exists up to equivalence a unique S(5,6,12) Steiner system '''W<sub>12</sub>''', and the group M<sub>12</sub> is its automorphism group. The subgroup M<sub>11</sub> is the stabilizer of a | Similarly, there exists up to equivalence a unique S(5,6,12) Steiner system '''W<sub>12</sub>''', and the group M<sub>12</sub> is its automorphism group. The subgroup M<sub>11</sub> is the stabilizer of a point. | ||
W<sub>12</sub> can be constructed from the [[affine geometry]] on the [[vector space]] F<sub>3</sub>xF<sub>3</sub>, an S(2,3,9) system. | W<sub>12</sub> can be constructed from the [[affine geometry]] on the [[vector space]] F<sub>3</sub>xF<sub>3</sub>, an S(2,3,9) system. | ||
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An introduction to a construction of W<sub>24</sub> via the [[Miracle Octad Generator]] of R. T. Curtis and Conway's analog for W<sub>12</sub>, the miniMOG, can be found in the book by Conway and [[Neil Sloane|Sloane]]. | An introduction to a construction of W<sub>24</sub> via the [[Miracle Octad Generator]] of R. T. Curtis and Conway's analog for W<sub>12</sub>, the miniMOG, can be found in the book by Conway and [[Neil Sloane|Sloane]]. | ||
=== Automorphism | === Automorphism groups on the Golay code === | ||
The group M<sub>24</sub> is the [[Mathieu_group_M24#Automorphism_group_of_the_Golay_code|permutation automorphism group]] of the [[binary Golay code|extended binary Golay code]] ''W'', i.e., the group of permutations on the 24 coordinates that map ''W'' to itself. All the Mathieu groups can be constructed as groups of permutations on the binary Golay code. | |||
The group M<sub>24</sub> | |||
M<sub>12</sub> has index 2 in its automorphism group | M<sub>12</sub> has index 2 in its automorphism group, and M<sub>12</sub>:2 happens to be isomorphic to a subgroup of M<sub>24</sub>. M<sub>12</sub> is the stabilizer of a '''dodecad''', a codeword of 12 1's; M<sub>12</sub>:2 stabilizes a partition into 2 complementary dodecads. | ||
There is a natural connection between the Mathieu groups and the larger [[Conway groups]], because | There is a natural connection between the Mathieu groups and the larger [[Conway groups]], because the [[Leech lattice]] was constructed on the binary Golay code and in fact both lie in spaces of dimension 24. The Conway groups in turn are found in the [[Monster group]]. [[Robert Griess]] refers to the 20 sporadic groups found in the Monster as the '''Happy Family''', and to the Mathieu groups as the '''first generation'''. | ||
===Dessins d'enfants=== | ===Dessins d'enfants=== | ||
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*{{Citation | last1=Cameron | first1=Peter J. | title=Permutation Groups | publisher=[[Cambridge University Press]] | series=London Mathematical Society Student Texts | isbn=978-0-521-65378-7 | year=1999 | volume=45}} | *{{Citation | last1=Cameron | first1=Peter J. | title=Permutation Groups | publisher=[[Cambridge University Press]] | series=London Mathematical Society Student Texts | isbn=978-0-521-65378-7 | year=1999 | volume=45}} | ||
*{{Citation | last1=Carmichael | first1=Robert D. | title=Introduction to the theory of groups of finite order | origyear=1937 | url=http://books.google.com/books?id=McMgAAAAMAAJ | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-60300-1 | | *{{Citation | last1=Carmichael | first1=Robert D. | title=Introduction to the theory of groups of finite order | origyear=1937 | url=http://books.google.com/books?id=McMgAAAAMAAJ | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-60300-1 |mr=0075938 | year=1956}} | ||
* {{cite jstor|1996123}} | * {{cite jstor|1996123}} | ||
* {{cite jstor|1996124}} | * {{cite jstor|1996124}} | ||
*{{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | editor1-last=Powell | editor1-first=M. B. | editor2-last=Higman | editor2-first=Graham | editor2-link=Graham Higman | title=Finite simple groups | url=http://books.google.com/books?id=TPPkAAAAIAAJ | publisher=[[Academic Press]] | location=Boston, MA | series=Proceedings of an Instructional Conference organized by the London Mathematical Society (a NATO Advanced Study Institute), Oxford, September 1969. | isbn=978-0-12-563850-0 | | *{{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | editor1-last=Powell | editor1-first=M. B. | editor2-last=Higman | editor2-first=Graham | editor2-link=Graham Higman | title=Finite simple groups | url=http://books.google.com/books?id=TPPkAAAAIAAJ | publisher=[[Academic Press]] | location=Boston, MA | series=Proceedings of an Instructional Conference organized by the London Mathematical Society (a NATO Advanced Study Institute), Oxford, September 1969. | isbn=978-0-12-563850-0 |mr=0338152 | year=1971 | chapter=Three lectures on exceptional groups | pages=215–247}} Reprinted in {{harvtxt|Conway|Sloane|1999|loc= 267–298}} | ||
*{{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | last2=Parker | first2=Richard A. | last3=Norton | first3=Simon P. | last4=Curtis | first4=R. T. | last5=Wilson | first5=Robert A. | title= | *{{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | last2=Parker | first2=Richard A. | last3=Norton | first3=Simon P. | last4=Curtis | first4=R. T. | last5=Wilson | first5=Robert A. | title=Atlas of finite groups | url=http://books.google.com/books?id=38fEMl2-Fp8C | publisher=[[Oxford University Press]] | isbn=978-0-19-853199-9 |mr=827219 | year=1985}} | ||
*{{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | last2=Sloane | first2=Neil J. A. | author2-link=Neil Sloane | title=Sphere Packings, Lattices and Groups | url=http://books.google.com/books?id=upYwZ6cQumoC | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=3rd | series=Grundlehren der Mathematischen Wissenschaften | isbn=978-0-387-98585-5 | | *{{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | last2=Sloane | first2=Neil J. A. | author2-link=Neil Sloane | title=Sphere Packings, Lattices and Groups | url=http://books.google.com/books?id=upYwZ6cQumoC | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=3rd | series=Grundlehren der Mathematischen Wissenschaften | isbn=978-0-387-98585-5 |mr=0920369 | year=1999 | volume=290}} | ||
*{{Citation | last1=Curtis | first1=R. T. | title=A new combinatorial approach to M₂₄ | doi=10.1017/S0305004100052075 | | *{{Citation | last1=Curtis | first1=R. T. | title=A new combinatorial approach to M₂₄ | doi=10.1017/S0305004100052075 |mr=0399247 | year=1976 | journal=Mathematical Proceedings of the Cambridge Philosophical Society | issn=0305-0041 | volume=79 | issue=1 | pages=25–42}} | ||
*{{Citation | last1=Curtis | first1=R. T. | title=The maximal subgroups of M₂₄ | doi=10.1017/S0305004100053251 | | *{{Citation | last1=Curtis | first1=R. T. | title=The maximal subgroups of M₂₄ | doi=10.1017/S0305004100053251 |mr=0439926 | year=1977 | journal=Mathematical Proceedings of the Cambridge Philosophical Society | issn=0305-0041 | volume=81 | issue=2 | pages=185–192}} | ||
*{{Citation | last1=Curtis | first1=R. T. | editor1-last=Atkinson | editor1-first=Michael D. | title=Computational group theory. Proceedings of the London Mathematical Society symposium held in Durham, July 30–August 9, 1982. | url=http://books.google.com/books?id=RvvuAAAAMAAJ | publisher=[[Academic Press]] | location=Boston, MA | isbn=978-0-12-066270-8 | | *{{Citation | last1=Curtis | first1=R. T. | editor1-last=Atkinson | editor1-first=Michael D. | title=Computational group theory. Proceedings of the London Mathematical Society symposium held in Durham, July 30–August 9, 1982. | url=http://books.google.com/books?id=RvvuAAAAMAAJ | publisher=[[Academic Press]] | location=Boston, MA | isbn=978-0-12-066270-8 |mr=760669 | year=1984 | chapter=The Steiner system S(5, 6, 12), the Mathieu group M₁₂ and the "kitten" | pages=353–358}} | ||
* {{Citation | * {{Citation | ||
|title=The Mathieu groups and their geometries | |title=The Mathieu groups and their geometries | ||
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|url=http://www.win.tue.nl/~hansc/mathieu.pdf | |url=http://www.win.tue.nl/~hansc/mathieu.pdf | ||
}} | }} | ||
*{{Citation | last1=Dixon | first1=John D. | last2=Mortimer | first2=Brian | title=Permutation groups | url=http://dx.doi.org/10.1007/978-1-4612-0731-3 | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-94599-6 | doi=10.1007/978-1-4612-0731-3 | | *{{Citation | last1=Dixon | first1=John D. | last2=Mortimer | first2=Brian | title=Permutation groups | url=http://dx.doi.org/10.1007/978-1-4612-0731-3 | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-94599-6 | doi=10.1007/978-1-4612-0731-3 |mr=1409812 | year=1996 | volume=163}} | ||
*{{Citation | last1=Frobenius | first1=Ferdinand Georg | author1-link=Ferdinand Georg Frobenius | title=Über die Charaktere der mehrfach transitiven Gruppen | url=http://books.google.com/books?id=ksNjpwAACAAJ | publisher=Mouton De Gruyter | series=Berline Berichte | isbn=978-3-11-109790-9 | year=1904 | pages=558–571}} | *{{Citation | last1=Frobenius | first1=Ferdinand Georg | author1-link=Ferdinand Georg Frobenius | title=Über die Charaktere der mehrfach transitiven Gruppen | url=http://books.google.com/books?id=ksNjpwAACAAJ | publisher=Mouton De Gruyter | series=Berline Berichte | isbn=978-3-11-109790-9 | year=1904 | pages=558–571}} | ||
*{{Citation | last1=Griess | first1=Robert L. Jr. | author1-link=R. L. Griess | title=Twelve sporadic groups | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-540-62778-4 | | *{{Citation | last1=Griess | first1=Robert L. Jr. | author1-link=R. L. Griess | title=Twelve sporadic groups | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-540-62778-4 |mr=1707296 | year=1998}} | ||
* {{Citation | last1=Mathieu | first1=Émile | title=Mémoire sur l'étude des fonctions de plusieurs quantités, sur la manière de les former et sur les substitutions qui les laissent invariables | url=http://gallica.bnf.fr/ark:/12148/bpt6k16405f/f249 | year=1861 | journal=Journal de Mathématiques Pures et Appliquées | volume=6 | pages=241–323}} | * {{Citation | last1=Mathieu | first1=Émile | title=Mémoire sur l'étude des fonctions de plusieurs quantités, sur la manière de les former et sur les substitutions qui les laissent invariables | url=http://gallica.bnf.fr/ark:/12148/bpt6k16405f/f249 | year=1861 | journal=Journal de Mathématiques Pures et Appliquées | volume=6 | pages=241–323}} | ||
*{{Citation | last1=Mathieu | first1=Émile | title=Sur la fonction cinq fois transitive de 24 quantités | url=http://portail.mathdoc.fr/JMPA/afficher_notice.php?id=JMPA_1873_2_18_A2_0 | language=French | | *{{Citation | last1=Mathieu | first1=Émile | title=Sur la fonction cinq fois transitive de 24 quantités | url=http://portail.mathdoc.fr/JMPA/afficher_notice.php?id=JMPA_1873_2_18_A2_0 | language=French |jfm=05.0088.01 | year=1873 | journal=Journal de Mathématiques Pures et Appliquées | volume=18 | pages=25–46}} | ||
*{{Citation | last1=Miller | first1=G. A. | title=On the supposed five-fold transitive function of 24 elements and 19!/48 values. | url=http://books.google.com/books?id=LMAKAAAAIAAJ&pg=PA187 | year=1898 | journal=[[Messenger of Mathematics]] | volume=27 | pages=187–190}} | *{{Citation | last1=Miller | first1=G. A. | title=On the supposed five-fold transitive function of 24 elements and 19!/48 values. | url=http://books.google.com/books?id=LMAKAAAAIAAJ&pg=PA187 | year=1898 | journal=[[Messenger of Mathematics]] | volume=27 | pages=187–190}} | ||
*{{Citation | last1=Miller | first1=G. A. | title=Sur plusieurs groupes simples | url=http://www.numdam.org/item?id=BSMF_1900__28__266_0 | year=1900 | journal= Bulletin de la Société Mathématique de France | volume=28 | pages=266–267}} | *{{Citation | last1=Miller | first1=G. A. | title=Sur plusieurs groupes simples | url=http://www.numdam.org/item?id=BSMF_1900__28__266_0 | year=1900 | journal= Bulletin de la Société Mathématique de France | volume=28 | pages=266–267}} | ||
*{{Citation | last1=Ronan | first1=Mark | title=Symmetry and the Monster | publisher=Oxford | isbn=978-0-19-280722-9 | year=2006}} (an introduction for the lay reader, describing the Mathieu groups in a historical context) | *{{Citation | last1=Ronan | first1=Mark | title=Symmetry and the Monster | publisher=Oxford | isbn=978-0-19-280722-9 | year=2006}} (an introduction for the lay reader, describing the Mathieu groups in a historical context) | ||
*{{Citation | last1=Thompson | first1=Thomas M. | title=From error-correcting codes through sphere packings to simple groups | url=http://books.google.com/books?id=ggqxuG31B3cC | publisher=[[Mathematical Association of America]] | series=Carus Mathematical Monographs | isbn=978-0-88385-023-7 | | *{{Citation | last1=Thompson | first1=Thomas M. | title=From error-correcting codes through sphere packings to simple groups | url=http://books.google.com/books?id=ggqxuG31B3cC | publisher=[[Mathematical Association of America]] | series=Carus Mathematical Monographs | isbn=978-0-88385-023-7 |mr=749038 | year=1983 | volume=21}} | ||
*{{Citation | last1=Witt | first1=Ernst | author1-link=Ernst Witt | title=über Steinersche Systeme | publisher=Springer Berlin / Heidelberg | doi=10.1007/BF02948948 | year=1938a | journal=Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg | issn=0025-5858 | volume=12 | pages=265–275}} | *{{Citation | last1=Witt | first1=Ernst | author1-link=Ernst Witt | title=über Steinersche Systeme | publisher=Springer Berlin / Heidelberg | doi=10.1007/BF02948948 | year=1938a | journal=Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg | issn=0025-5858 | volume=12 | pages=265–275}} | ||
*{{Citation | last1=Witt | first1=Ernst | author1-link=Ernst Witt | title=Die 5-fach transitiven Gruppen von Mathieu | doi=10.1007/BF02948947 | year=1938b | journal=Abhandlungen | *{{Citation | last1=Witt | first1=Ernst | author1-link=Ernst Witt | title=Die 5-fach transitiven Gruppen von Mathieu | doi=10.1007/BF02948947 | year=1938b | journal=[[Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg]] | volume=12 | pages=256–264}} | ||
==External links== | ==External links== | ||
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{{DEFAULTSORT:Mathieu Group}} | {{DEFAULTSORT:Mathieu Group}} | ||
[[Category:Sporadic groups]] | [[Category:Sporadic groups]] | ||
Latest revision as of 09:34, 25 November 2014
In mathematics, the Mathieu groups M_{11}, M_{12}, M_{22}, M_{23}, M_{24}, introduced by Template:Harvs, are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objects. They were the first sporadic simple groups discovered.
Sometimes the notation M_{10}, M_{20} and M_{21} is used for related groups (which act on sets of 10, 20, and 21 points, respectively), namely the stabilizers of points in the larger groups. While these are not sporadic simple groups, they are subgroups of the larger groups and can be used to construct the larger ones. John Conway has shown that one can also extend this sequence up, obtaining the Mathieu groupoid M_{13} acting on 13 points. M_{21} is simple, but is not a sporadic group, being isomorphic to PSL(3,4).
History
Template:Harvtxt introduced the group M_{12} as part of an investigation of multiply transitive permutation groups, and briefly mentioned (on page 274) the group M_{24}, giving its order. In Template:Harvtxt he gave further details, including explicit generating sets for his groups, but it was not easy to see from his arguments that the groups generated are not just alternating groups, and for several years the existence of his groups was controversial. Template:Harvtxt even published a paper mistakenly claiming to prove that M_{24} does not exist, though shortly afterwards in Template:Harv he pointed out that his proof was wrong, and gave a proof that the Mathieu groups are simple. Template:Harvs finally removed the doubts about the existence of these groups, by constructing them as automorphism groups of Steiner systems.
After the Mathieu groups no new sporadic groups were found until 1965, when the group J_{1} was discovered.
Multiply transitive groups
Mathieu was interested in finding multiply transitive permutation groups, which will now be defined. For a natural number k, a permutation group G acting on n points is k-transitive if, given two sets of points a_{1}, ... a_{k} and b_{1}, ... b_{k} with the property that all the a_{i} are distinct and all the b_{i} are distinct, there is a group element g in G which maps a_{i} to b_{i} for each i between 1 and k. Such a group is called sharply k-transitive if the element g is unique (i.e. the action on k-tuples is regular, rather than just transitive).
M_{24} is 5-transitive, and M_{12} is sharply 5-transitive, with the other Mathieu groups (simple or not) being the subgroups corresponding to stabilizers of m points, and accordingly of lower transitivity (M_{23} is 4-transitive, etc.).
The only 4-transitive groups are the symmetric groups S_{k} for k at least 4, the alternating groups A_{k} for k at least 6, and the Mathieu groups M_{24}, M_{23}, M_{12} and M_{11}. Template:Harv The full proof requires the classification of finite simple groups, but some special cases have been known for much longer.
It is a classical result of Jordan that the symmetric and alternating groups (of degree k and k + 2 respectively), and M_{12} and M_{11} are the only sharply k-transitive permutation groups for k at least 4.
Important examples of multiply transitive groups are the 2-transitive groups and the Zassenhaus groups. The Zassenhaus groups notably include the projective general linear group of a projective line over a finite field, PGL(2,F_{q}), which is sharply 3-transitive (see cross ratio) on elements.
Order and transitivity table
Group | Order | Order (product) | Factorised order | Transitivity | Simple |
---|---|---|---|---|---|
M_{24} | 244823040 | 3·16·20·21·22·23·24 | 2^{10}·3^{3}·5·7·11·23 | 5-transitive | simple |
M_{23} | 10200960 | 3·16·20·21·22·23 | 2^{7}·3^{2}·5·7·11·23 | 4-transitive | simple |
M_{22} | 443520 | 3·16·20·21·22 | 2^{7}·3^{2}·5·7·11 | 3-transitive | simple |
M_{21} | 20160 | 3·16·20·21 | 2^{6}·3^{2}·5·7 | 2-transitive | simple |
M_{20} | 960 | 3·16·20 | 2^{6}·3·5 | 1-transitive | not simple |
M_{12} | 95040 | 8·9·10·11·12 | 2^{6}·3^{3}·5·11 | sharply 5-transitive | simple |
M_{11} | 7920 | 8·9·10·11 | 2^{4}·3^{2}·5·11 | sharply 4-transitive | simple |
M_{10} | 720 | 8·9·10 | 2^{4}·3^{2}·5 | sharply 3-transitive | not simple |
Constructions of the Mathieu groups
The Mathieu groups can be constructed in various ways.
Permutation groups
M_{12} has a simple subgroup of order 660, a maximal subgroup. That subgroup can be represented as a linear fractional group on the field F_{11} of 11 elements. With −1 written as a and infinity as b, two standard generators are (0123456789a) and (0b)(1a)(25)(37)(48)(69). A third generator giving M_{12} sends an element x of F_{11} to 4x^{2} − 3x^{7}; as a permutation that is (26a7)(3945). The stabilizer of 4 points is a quaternion group.
Likewise M_{24} has a maximal simple subgroup of order 6072 and this can be represented as a linear fractional group on the field F_{23}. One generator adds 1 to each element (leaving the point N at infinity fixed), i. e. (0123456789ABCDEFGHIJKLM)(N), and the other is the order reversing permutation, (0N)(1M)(2B)(3F)(4H)(59)(6J)(7D)(8K)(AG)(CL)(EI). A third generator giving M_{24} sends an element x of F_{23} to 4x^{4} − 3x^{15} (which sends perfect squares via and non-perfect squares via ); computation shows that as a permutation this is (2G968)(3CDI4)(7HABM)(EJLKF).
These constructions were cited by Template:Harvtxt. Template:Harvtxt ascribe the permutations to Mathieu.
Automorphism groups of Steiner systems
There exists up to equivalence a unique S(5,8,24) Steiner system W_{24} (the Witt design). The group M_{24} is the automorphism group of this Steiner system; that is, the set of permutations which map every block to some other block. The subgroups M_{23} and M_{22} are defined to be the stabilizers of a single point and two points respectively.
Similarly, there exists up to equivalence a unique S(5,6,12) Steiner system W_{12}, and the group M_{12} is its automorphism group. The subgroup M_{11} is the stabilizer of a point.
W_{12} can be constructed from the affine geometry on the vector space F_{3}xF_{3}, an S(2,3,9) system.
An alternative construction of W_{12} is the 'Kitten' of Template:Harvtxt.
An introduction to a construction of W_{24} via the Miracle Octad Generator of R. T. Curtis and Conway's analog for W_{12}, the miniMOG, can be found in the book by Conway and Sloane.
Automorphism groups on the Golay code
The group M_{24} is the permutation automorphism group of the extended binary Golay code W, i.e., the group of permutations on the 24 coordinates that map W to itself. All the Mathieu groups can be constructed as groups of permutations on the binary Golay code.
M_{12} has index 2 in its automorphism group, and M_{12}:2 happens to be isomorphic to a subgroup of M_{24}. M_{12} is the stabilizer of a dodecad, a codeword of 12 1's; M_{12}:2 stabilizes a partition into 2 complementary dodecads.
There is a natural connection between the Mathieu groups and the larger Conway groups, because the Leech lattice was constructed on the binary Golay code and in fact both lie in spaces of dimension 24. The Conway groups in turn are found in the Monster group. Robert Griess refers to the 20 sporadic groups found in the Monster as the Happy Family, and to the Mathieu groups as the first generation.
Dessins d'enfants
The Mathieu groups can be constructed via dessins d'enfants, with the dessin associated to M_{12} suggestively called "Monsieur Mathieu" by Template:Harvtxt.
References
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External links
- ATLAS: Mathieu group M_{10}
- ATLAS: Mathieu group M_{11}
- ATLAS: Mathieu group M_{12}
- ATLAS: Mathieu group M_{20}
- ATLAS: Mathieu group M_{21}
- ATLAS: Mathieu group M_{22}
- ATLAS: Mathieu group M_{23}
- ATLAS: Mathieu group M_{24}
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- Scientific American A set of puzzles based on the mathematics of the Mathieu groups
- Sporadic M12 An iPhone app that implements puzzles based on M_{12}, presented as one "spin" permutation and a selectable "swap" permutation