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| {{DISPLAYTITLE:Janko group J<sub>4</sub>}}
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| {{Group theory sidebar |Finite}}
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| In [[mathematics]], the '''fourth Janko group''' ''J''<sub>4</sub> is the [[sporadic group|sporadic]] [[finite simple group]] of order
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| :2<sup>21</sup>{{·}}3<sup>3</sup>{{·}}5{{·}}7{{·}}11<sup>3</sup>{{·}}23{{·}}29{{·}}31{{·}}37{{·}}43 (= 86775571046077562880)
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| whose existence was suggested by {{harvs|txt |authorlink=Zvonimir Janko |last=Janko |year=1976}}.
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| The [[Schur multiplier]] and the [[outer automorphism group]] are both [[Triviality (mathematics)|trivial]].
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| J<sub>4</sub> is one of the 6 sporadic simple groups known as the "[[pariah group]]s" as they are not found within the [[Monster group]]. The order of the monster group is not divisible by 37 or 43.
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| == Existence and uniqueness ==
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| Janko found J₄ by studying groups with an involution centralizer of the form 2<sup>1 + 12</sup>.3.(M<sub>22</sub>:2).
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| Its existence and uniqueness was shown using computer calculations by [[Simon P. Norton]] and others in 1980. It has a [[modular representation]] of dimension 112 over the finite field of two elements and is the stabilizer of a certain 4995 dimensional subspace of the exterior square, a fact which Norton used to construct it, and which is the easiest way to deal with it computationally. {{harvtxt|Aschbacher|Segev|1991}} and {{harvtxt|Ivanov|1992}} gave computer-free proofs of uniqueness. {{harvtxt|Ivanov|Meierfrankenfeld|1999}} and {{harvtxt|Ivanov|2004}} gave a computer-free proof of existence by constructing it as an amalgams of groups 2<sup>10</sup>:SL<sub>5</sub>(2) and (2<sup>10</sup>:2<sup>4</sup>:A<sub>8</sub>):2 over a group 2<sup>10</sup>:2<sup>4</sup>:A<sub>8</sup>.
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| ==Representations==
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| The smallest faithful complex representation has dimension 1333; there are two complex conjugate representations of this dimension. The smallest faithful representation over any field is a 112 dimensional representation over the field of 2 elements.
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| The smallest permutation representation is on 173067389 points, with point stabilizer of the form 2<sup>11</sup>M<sub>24</sub>. These points can be identified with certain "special vectors" in the 112 dimensional representation.
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| ==Presentation==
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| It has a presentation in terms of three generators a, b, and c as
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| :<math>a^2=b^3=c^2=(ab)^{23}=[a,b]^{12}=[a,bab]^5=[c,a]=</math>
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| :<math>(ababab^{-1})^3(abab^{-1}ab^{-1})^3=(ab(abab^{-1})^3)^4=</math>
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| :<math>[c,bab(ab^{-1})^2(ab)^3]=(bc^{bab^{-1}abab^{-1}a})^3=</math>
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| :<math>((bababab)^3cc^{(ab)^3b(ab)^6b})^2=1.</math>
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| ==Maximal subgroups==
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| {{harvtxt|Kleidman|Wilson|1988}} showed that J<sub>4</sub> has 13 conjugacy classes of maximal subgroups.
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| * 2<sup>11</sup>:M<sub>24</sub> - containing Sylow 2-subgroups and Sylow 3-subgroups; also containing 2<sup>11</sup>:(M<sub>22</sub>:2), centralizer of involution of class 2B
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| * 2<sup>1+12</sup>.3.(M<sub>22</sub>:2) - centralizer of involution of class 2A - containing Sylow 2-subgroups and Sylow 3-subgroups
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| * 2<sup>10</sup>:PSL(5,2)
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| * 2<sup>3+12</sup>.(S<sub>5</sub> × PSL(3,2)) - containing Sylow 2-subgroups
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| * U<sub>3</sub>(11):2
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| * M<sub>22</sub>:2
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| * 11<sup>1+2</sup>:(5 × GL(2,3)) - normalizer of Sylow 11-subgroup
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| * PSL(2,32):5
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| * PGL(2,23)
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| * U<sub>3</sub>(3) - containing Sylow 3-subgroups
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| * 29:28 Frobenius group
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| * 43:14 Frobenius group
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| * 37:12 Frobenius group
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| A Sylow 3-subgroup is a [[Heisenberg group]]: order 27, non-abelian, all non-trivial elements of order 3
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| == References ==
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| *{{Citation | last1=Aschbacher | first1=Michael | author1-link=Michael Aschbacher | last2=Segev | first2=Yoav | title=The uniqueness of groups of type J₄ | url=http://dx.doi.org/10.1007/BF01232280 | doi=10.1007/BF01232280 | id={{MR|1117152}} | year=1991 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=105 | issue=3 | pages=589–607}}
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| *D.J. Benson ''The simple group J<sub>4</sub>'', PhD Thesis, Cambridge 1981, http://www.maths.abdn.ac.uk/~bensondj/papers/b/benson/the-simple-group-J4.pdf
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| *{{Citation | last1=Ivanov | first1=A. A. | title=A presentation for J₄ | url=http://dx.doi.org/10.1112/plms/s3-64.2.369 | doi=10.1112/plms/s3-64.2.369 | id={{MR|1143229}} | year=1992 | journal=Proceedings of the London Mathematical Society. Third Series | issn=0024-6115 | volume=64 | issue=2 | pages=369–396}}
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| *{{Citation | last1=Ivanov | first1=A. A. | last2=Meierfrankenfeld | first2=Ulrich | title=A computer-free construction of J₄ | url=http://dx.doi.org/10.1006/jabr.1999.7851 | doi=10.1006/jabr.1999.7851 | id={{MR|1707666}} | year=1999 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=219 | issue=1 | pages=113–172}}
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| *Ivanov, A. A. ''The fourth Janko group.'' Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2004. xvi+233 pp. ISBN 0-19-852759-4 {{MR|2124803}}
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| *Z. Janko, ''A new finite simple group of order 86,775,570,046,077,562,880 which possesses M<sub>24</sub> and the full covering group of M<sub>22</sub> as subgroups'', J. Algebra 42 (1976) 564-596.{{DOI|10.1016/0021-8693(76)90115-0}} (The title of this paper is incorrect, as the full covering group of M<sub>22</sub> was later discovered to be larger: center of order 12, not 6.)
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| *{{Citation | last1=Kleidman | first1=Peter B. | last2=Wilson | first2=Robert A. | title=The maximal subgroups of J<sub>4</sub> | doi=10.1112/plms/s3-56.3.484 | mr=931511 | year=1988 | journal=Proceedings of the London Mathematical Society. Third Series | issn=0024-6115 | volume=56 | issue=3 | pages=484–510}}
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| *S. P. Norton ''The construction of J<sub>4</sub>'' in ''The Santa Cruz conference on finite groups'' (Ed. Cooperstein, Mason) Amer. Math. Soc 1980.
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| ==External links==
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| * [http://web.mat.bham.ac.uk/atlas/v2.0/spor/J4/ Atlas of Finite Group Representations version 2: ''J''<sub>4</sub>]
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| *[http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/J4/ Atlas of Finite Group Representations version 3: ''J''<sub>4</sub>]
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| [[Category:Sporadic groups]]
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