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| {{Group theory sidebar |Finite}}
| | Our world is driven by existing plus demand. Many shall examine the Greek-Roman model. Consuming additional care to highlight the aspect of [http://prometeu.net clash of clans hack tool] no evaluation within the vast system which usually this gives you.<br><br>Trying the higher level: it is essential when it comes with a game, but when it depends on Clash of Clans, have a lot more subtle simple steps. Despite making use of clash of clans hack tools, you may acquire experience points through the process of matching on top linked to other players. Lessen purpose of earning Pvp is to enable further enhancements for your indigneous group. The improvement consists of better fight equipment, properties, troops while tribe people.<br><br>To take pleasure from unlimited points, resources, gold and silver coins or gems, you must download the clash of clans crack tool by clicking to your button. Depending about the operating system that an individual using, you will will need to run the downloaded list as [http://search.Huffingtonpost.com/search?q=administrator&s_it=header_form_v1 administrator]. Necessary under some log in ID and select the device. Immediately this, you are ought enter the number together with gems or coins you require to get.<br><br>Whether or not you are searching towards a particular game into buy but want which can purchase it at their best price possible, exercise the "shopping" tab purchasable on many search search engines. This will allocate you to immediately consider the prices of this particular game at all the major retailers online. You can also read ratings for the trader in question, helping your [http://www.reddit.com/r/howto/search?q=business+determine business determine] who you want to buy the game because of.<br><br>Everybody true, you've landed in the correct spot! Truly, we have produced instantly lengthy hrs of research, perform and screening, a response for thr Clash amongst Clans Cheat totally unknown and operates perfectly. And due to the trouble of our teams, your company never-ending hrs of enjoyment in your iPhone, ipad booklet or iPod Touch performing Clash of Clans the cheat code Clash having to do with Clans produced especially to aid you!<br><br>You will see for yourself that this useful Money Compromise of Clans i fairly effective, really invisible by the manager of the game, predominantly absolutely no price!<br><br>To allow them to conclude, clash of clans hack tool no piece of research must not be approved to get in method of the bigger question: what makes we to this article? Putting this aside the truck bed cover's of great importance. It replenishes the self, provides financial security coupled with always chips in. |
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| 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.
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| 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.
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| == History ==
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| {{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.
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| 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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| == Multiply transitive groups ==
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| 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''<sub>1</sub>, ... ''a''<sub>''k''</sub> and ''b''<sub>1</sub>, ... ''b''<sub>''k''</sub> with the property that all the ''a''<sub>''i''</sub> are distinct and all the ''b''<sub>''i''</sub> are distinct, there is a group element ''g'' in ''G'' which maps ''a''<sub>''i''</sub> to ''b''<sub>''i''</sub> 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 [[group action#Types of actions|regular]], rather than just transitive).
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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.).
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| 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.
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| 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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| Important examples of multiply transitive groups are the [[2-transitive group]]s and the [[Zassenhaus group]]s. The Zassenhaus groups notably include the [[projective general linear group]] of a projective line over a finite field, PGL(2,'''F'''<sub>''q''</sub>), which is sharply 3-transitive (see [[cross ratio]]) on <math>q+1</math> elements.
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| === Order and transitivity table === | |
| {| class="wikitable"
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| ! Group
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| ! Order
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| ! Order (product)
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| ! Factorised order
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| ! Transitivity
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| ! Simple
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| |-
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| | M<sub>24</sub>
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| | 244823040
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| | 3·16·20·21·22·23·24
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| | 2<sup>10</sup>·3<sup>3</sup>·5·7·11·23
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| | 5-transitive
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| | simple
| |
| |-
| |
| | M<sub>23</sub>
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| | 10200960
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| | 3·16·20·21·22·23
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| | 2<sup>7</sup>·3<sup>2</sup>·5·7·11·23
| |
| | 4-transitive
| |
| | simple
| |
| |-
| |
| | M<sub>22</sub>
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| | 443520
| |
| | 3·16·20·21·22
| |
| | 2<sup>7</sup>·3<sup>2</sup>·5·7·11
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| | 3-transitive
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| | simple
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| |-
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| | M<sub>21</sub>
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| | 20160
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| | 3·16·20·21
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| | 2<sup>6</sup>·3<sup>2</sup>·5·7
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| | 2-transitive
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| | simple
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| |-
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| | M<sub>20</sub>
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| | 960
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| | 3·16·20
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| | 2<sup>6</sup>·3·5
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| | 1-transitive
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| | not simple
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| |-
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| | colspan = "5" |
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| |-
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| | M<sub>12</sub>
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| | 95040
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| | 8·9·10·11·12
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| | 2<sup>6</sup>·3<sup>3</sup>·5·11
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| | sharply 5-transitive
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| | simple
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| |-
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| | M<sub>11</sub>
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| | 7920
| |
| | 8·9·10·11
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| | 2<sup>4</sup>·3<sup>2</sup>·5·11
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| | sharply 4-transitive
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| | simple
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| |-
| |
| | M<sub>10</sub>
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| | 720
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| | 8·9·10
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| | 2<sup>4</sup>·3<sup>2</sup>·5
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| | sharply 3-transitive
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| | not simple
| |
| |}
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| == Constructions of the Mathieu groups ==
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| The Mathieu groups can be constructed in various ways.
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| ===Permutation groups===
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| 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]].
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| 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).
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| 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.
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| === 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.
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| 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.
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| 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 alternative construction of W<sub>12</sub> is the 'Kitten' of {{Harvtxt|Curtis|1984}}.
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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]].
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| === Automorphism group of the Golay code ===
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| The group M<sub>24</sub> also is the permutation [[automorphism group]] of the [[binary Golay code]] ''W'', i.e., the group of permutations of coordinates mapping ''W'' to itself. (In coding theory the term "binary Golay code" often refers to a shorter related length 23 code, and the length 24 code used here is called the "extended binary Golay code".) Codewords correspond in a natural way to subsets of a set of 24 objects. Those subsets corresponding to codewords with 8 or 12 coordinates equal to 1 are called '''octads''' or '''dodecads''' respectively. The octads are the blocks of an S(5,8,24) Steiner system and the binary Golay code is the vector space over field F<sub>2</sub> spanned by the octads of the Steiner system.
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| The simple subgroups M<sub>23</sub>, M<sub>22</sub>, M<sub>12</sub>, and M<sub>11</sub> can be defined as subgroups of M<sub>24</sub>, stabilizers respectively of a single coordinate, an ordered pair of coordinates, a dodecad, and a dodecad together with a single coordinate.
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| M<sub>12</sub> has index 2 in its automorphism group. As a subgroup of M<sub>24</sub>, M<sub>12</sub> acts on the second dodecad as an outer automorphic image of its action on the first dodecad. M<sub>11</sub> is a subgroup of M<sub>23</sub> but not of M<sub>22</sub>. This representation of M<sub>11</sub> has orbits of 11 and 12. The automorphism group of M<sub>12</sub> is a maximal subgroup of M<sub>24</sub> of index 1288.
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| There is a natural connection between the Mathieu groups and the larger [[Conway groups]], because the binary Golay code and the [[Leech lattice]] 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'''.
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| ===Dessins d'enfants===
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| The Mathieu groups can be constructed via [[dessins d'enfants]], with the dessin associated to M<sub>12</sub> suggestively called "Monsieur Mathieu" by {{harvtxt|le Bruyn|2007}}.
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| == References ==
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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}}
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| *{{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 | id={{MR|0075938}} | year=1956}}
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| * {{cite jstor|1996123}}
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| * {{cite jstor|1996124}}
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| *{{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 | id={{MR|0338152}} | year=1971 | chapter=Three lectures on exceptional groups | pages=215–247}} Reprinted in {{harvtxt|Conway|Sloane|1999|loc= 267–298}}
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| *{{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 | id={{MathSciNet | id = 827219}} | year=1985}}
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| *{{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 | id={{MR|0920369}} | year=1999 | volume=290}}
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| *{{Citation | last1=Curtis | first1=R. T. | title=A new combinatorial approach to M₂₄ | doi=10.1017/S0305004100052075 | id={{MR|0399247}} | year=1976 | journal=Mathematical Proceedings of the Cambridge Philosophical Society | issn=0305-0041 | volume=79 | issue=1 | pages=25–42}}
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| *{{Citation | last1=Curtis | first1=R. T. | title=The maximal subgroups of M₂₄ | doi=10.1017/S0305004100053251 | id={{MR|0439926}} | year=1977 | journal=Mathematical Proceedings of the Cambridge Philosophical Society | issn=0305-0041 | volume=81 | issue=2 | pages=185–192}}
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| *{{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 | id={{MR|760669}} | year=1984 | chapter=The Steiner system S(5, 6, 12), the Mathieu group M₁₂ and the "kitten" | pages=353–358}}
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| * {{Citation
| |
| |title=The Mathieu groups and their geometries
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| |first=Hans
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| |last=Cuypers
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| |url=http://www.win.tue.nl/~hansc/mathieu.pdf
| |
| }}
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| *{{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 | id={{MR|1409812}} | year=1996 | volume=163}}
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| *{{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}}
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| *{{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 | id={{MathSciNet | id = 1707296}} | year=1998}}
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| * {{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}}
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| *{{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 | id={{JFM|05.0088.01}} | year=1873 | journal=Journal de Mathématiques Pures et Appliquées | volume=18 | pages=25–46}}
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| *{{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}}
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| *{{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}}
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| *{{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)
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| *{{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 | id={{MR|749038}} | year=1983 | volume=21}}
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| *{{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}}
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| *{{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}}
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| ==External links==
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| * [http://brauer.maths.qmul.ac.uk/Atlas/v3/group/M10/ ATLAS: Mathieu group M<sub>10</sub>]
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| * [http://brauer.maths.qmul.ac.uk/Atlas/v3/group/M11/ ATLAS: Mathieu group M<sub>11</sub>]
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| * [http://brauer.maths.qmul.ac.uk/Atlas/v3/group/M12/ ATLAS: Mathieu group M<sub>12</sub>]
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| * [http://brauer.maths.qmul.ac.uk/Atlas/v3/group/M20/ ATLAS: Mathieu group M<sub>20</sub>]
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| * [http://brauer.maths.qmul.ac.uk/Atlas/v3/group/M21/ ATLAS: Mathieu group M<sub>21</sub>]
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| * [http://brauer.maths.qmul.ac.uk/Atlas/v3/group/M22/ ATLAS: Mathieu group M<sub>22</sub>]
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| * [http://brauer.maths.qmul.ac.uk/Atlas/v3/group/M23/ ATLAS: Mathieu group M<sub>23</sub>]
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| * [http://brauer.maths.qmul.ac.uk/Atlas/v3/group/M24/ ATLAS: Mathieu group M<sub>24</sub>]
| |
| *{{citation
| |
| | last = le Bruyn | first = Lieven
| |
| | title = Monsieur Mathieu
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| | year = 2007
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| | url = http://www.neverendingbooks.org/index.php/monsieur-mathieu.html}}
| |
| * {{citation | ref = {{harvid|Richter}} | first = David A. | last = Richter | url = http://homepages.wmich.edu/~drichter/mathieu.htm | title = How to Make the Mathieu Group M<sub>24</sub> | accessdate = 2010-04-15 }}
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| {{refend}}
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| *[http://www.sciam.com/article.cfm?id=puzzles-simple-groups-at-play Scientific American] A set of puzzles based on the mathematics of the Mathieu groups
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| *[http://itunes.apple.com/us/app/sporadic-m12/id322438247 Sporadic M12 ] An iPhone app that implements puzzles based on M<sub>12</sub>, presented as one "spin" permutation and a selectable "swap" permutation
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| | |
| {{DEFAULTSORT:Mathieu Group}}
| |
| [[Category:Sporadic groups]]
| |
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