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| {{Group theory sidebar |Finite}}
| | Yoshiko is her name but she doesn't like when individuals use her complete title. The favorite pastime for my children and me is taking part in crochet and now I'm trying to make cash with it. Arizona is her birth location and she will never move. Bookkeeping is what I do for a residing.<br><br>Also visit my homepage ... extended car warranty ([http://chaddibaddi.com/blogs/post/24854 please click chaddibaddi.com]) |
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| In [[mathematics]], the ''Hall–Janko group'' ''HJ'', is a [[Finite group|finite]] [[Simple group|simple]] [[sporadic group]] of [[Order (group theory)|order]] 604800. It is also called the '''second Janko group''' ''J''<sub>2</sub>, or the '''Hall-Janko-Wales group''', since it was predicted by {{harvs|txt|authorlink=Zvonimir Janko|last=Janko|year=1969}} as one of two new groups with an involution centralizer of the form 2<sup>1+4</sup>A<sub>5</sub> (the other is the [[Janko group J3]]) and constructed by {{harvs|txt |authorlink=Marshall Hall (mathematician) |last1=Hall |last2=Wales |year=1968}} as a [[rank 3 permutation group]] on 100 points.
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| J<sub>2</sub> is the only one of the 4 [[Janko group]]s that is a section of the [[Monster group]]; it is thus part of what [[Robert Griess]] calls the Happy Family. Since it is also found in the Conway group Co<sub>1</sub>, it is therefore part of the second generation of the Happy Family.
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| == Representations ==
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| It is a subgroup of [[Index of a subgroup|index]] two of the group of automorphisms of the [[Hall-Janko graph]], leading to a permutation representation of degree 100.
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| It has a [[modular representation]] of dimension six over the field of four elements; if in [[characteristic (algebra)|characteristic]] two we have
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| ''w''<sup>2</sup> + ''w'' + 1 = 0, then J<sub>2</sub> is generated by the two matrices
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| :<math>
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| {\mathbf A} = \left ( \begin{matrix}
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| w^2 & w^2 & 0 & 0 & 0 & 0 \\
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| 1 & w^2 & 0 & 0 & 0 & 0 \\
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| 1 & 1 & w^2 & w^2 & 0 & 0 \\
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| w & 1 & 1 & w^2 & 0 & 0 \\
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| 0 & w^2 & w^2 & w^2 & 0 & w \\
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| w^2 & 1 & w^2 & 0 & w^2 & 0 \end{matrix} \right )
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| </math>
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| and | |
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| :<math>
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| {\mathbf B} = \left ( \begin{matrix}
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| w & 1 & w^2 & 1 & w^2 & w^2 \\
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| w & 1 & w & 1 & 1 & w \\
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| w & w & w^2 & w^2 & 1 & 0 \\
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| 0 & 0 & 0 & 0 & 1 & 1 \\
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| w^2 & 1 & w^2 & w^2 & w & w^2 \\
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| w^2 & 1 & w^2 & w & w^2 & w \end{matrix} \right ).
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| </math>
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| These matrices satisfy the equations
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| :<math>
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| {\mathbf A}^2 = {\mathbf B}^3 = ({\mathbf A}{\mathbf B})^7 =
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| ({\mathbf A}{\mathbf B}{\mathbf A}{\mathbf B}{\mathbf B})^{12} = 1.
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| </math>
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| J<sub>2</sub> is thus a [[Hurwitz group]], a finite homomorphic image of the [[(2,3,7) triangle group]].
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| The matrix representation given above constitutes an embedding into [[Leonard Eugene Dickson|Dickson's]] group [[Group of Lie type|''G''<sub>2</sub>(4)]]. There are two conjugacy classes of HJ in G<sub>2</sub>(4), and they are equivalent under the automorphism on the field F<sub>4</sub>. Their intersection (the "real" subgroup) is simple of order 6048. G<sub>2</sub>(4) is in turn isomorphic to a subgroup of the [[Conway group]] Co<sub>1</sub>.
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| ==Maximal subgroups==
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| J<sub>2</sub> has 9 conjugacy classes of [[maximal subgroup]]s. Some are here described in terms of action on the Hall-Janko graph.
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| * U<sub>3</sub>(3) order 6048 - one-point stabilizer, with orbits of 36 and 63
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| :Simple, containing 36 simple subgroups of order 168 and 63 involutions, all conjugate, each moving 80 points. A given involution is found in 12 168-subgroups, thus fixes them under conjugacy. Its centralizer has structure 4.S<sub>4</sub>, which contains 6 additional involutions. | |
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| * 3.PGL(2,9) order 2160 - has a subquotient A<sub>6</sub>
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| * 2<sup>1+4</sup>:A<sub>5</sub> order 1920 - centralizer of involution moving 80 points
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| * 2<sup>2+4</sup>:(3 × S<sub>3</sub>) order 1152
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| * A<sub>4</sub> × A<sub>5</sub> order 720
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| :Containing 2<sup>2</sup> × A<sub>5</sub> (order 240), centralizer of 3 involutions each moving 100 points
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| * A<sub>5</sub> × D<sub>10</sub> order 600
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| * PGL(2,7) order 336
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| * 5<sup>2</sup>:D<sub>12</sub> order 300
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| * A<sub>5</sub> order 60
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| ==Conjugacy classes==
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| The maximum order of any element is 15. As permutations, elements act on the 100 vertices of the Hall-Janko graph.
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| {| class="wikitable" style="margin: 1em auto;"
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| ! Order || No. elements || Cycle structure and conjugacy
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| | 1 = 1 || 1 = 1 || 1 class
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| |-
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| | rowspan="2" | 2 = 2 || 315 = 3<sup>2</sup> · 5 · 7 || 2<sup>40</sup>, 1 class
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| |-
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| | <!-- 2 = 2 ||--> 2520 = 2<sup>3</sup> · 3<sup>2</sup> · 5 · 7 || 2<sup>50</sup>, 1 class
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| |-
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| | rowspan="2" | 3 = 3 || 560 = 2<sup>4</sup> · 5 · 7 || 3<sup>30</sup>, 1 class
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| |-
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| | <!-- 3 = 3 ||--> 16800 = 2<sup>5</sup> · 3 · 5<sup>2</sup> · 7 || 3<sup>32</sup>, 1 class
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| |-
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| | 4 = 2<sup>2</sup> || 6300 = 2<sup>2</sup> · 3<sup>2</sup> · 5<sup>2</sup> · 7 || 2<sup>6</sup>4<sup>20</sup>, 1 class
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| |-
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| | rowspan="2" | 5 = 5 || 4032 = 2<sup>6</sup> · 3<sup>2</sup> · 7 || 5<sup>20</sup>, 2 classes, power equivalent
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| |-
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| | <!-- 5 = 5 ||--> 24192 = 2<sup>7</sup> · 3<sup>3</sup> · 7 || 5<sup>20</sup>, 2 classes, power equivalent
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| | rowspan="2" | 6 = 2 · 3 || 25200 = 2<sup>4</sup> · 3<sup>2</sup> · 5<sup>2</sup> · 7 || 2<sup>4</sup>3<sup>6</sup>6<sup>12</sup>, 1 class
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| |-
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| | <!-- 6 = 2 · 3 || --> 50400 = 2<sup>5</sup> · 3<sup>2</sup> · 5<sup>2</sup> · 7 || 2<sup>2</sup>6<sup>16</sup>, 1 class
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| | 7 = 7 || 86400 = 2<sup>7</sup> · 3<sup>3</sup> · 5<sup>2</sup> || 7<sup>14</sup>, 1 class
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| |-
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| | 8 = 2<sup>3</sup> || 75600 = 2<sup>4</sup> · 3<sup>3</sup> · 5<sup>2</sup> · 7 || 2<sup>3</sup>4<sup>3</sup>8<sup>10</sup>, 1 class
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| | rowspan="2" | 10 = 2 · 5 || 60480 = 2<sup>6</sup> · 3<sup>3</sup> · 5 · 7 || 10<sup>10</sup>, 2 classes, power equivalent
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| |-
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| | <!-- 10 = 2 · 5 ||--> 120960 = 2<sup>7</sup> · 3<sup>3</sup> · 5 · 7 || 5<sup>4</sup>10<sup>8</sup>, 2 classes, power equivalent
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| |-
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| | 12 = 2<sup>2</sup> · 3 || 50400 = 2<sup>5</sup> · 3<sup>2</sup> · 5<sup>2</sup> · 7 || 3<sup>2</sup>4<sup>2</sup>6<sup>2</sup>12<sup>6</sup>, 1 class
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| |-
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| | 15 = 3 · 5 || 80640 = 2<sup>8</sup> · 3<sup>2</sup> · 5 · 7 || 5<sup>2</sup>15<sup>6</sup>, 2 classes, power equivalent
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| |}
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| == References ==
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| * [[Robert L. Griess]], Jr., "Twelve Sporadic Groups", Springer-Verlag, 1998.
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| *{{Citation | last1=Hall | first1=Marshall | last2=Wales | first2=David | title=The simple group of order 604,800 | doi=10.1016/0021-8693(68)90014-8 | id={{MR|0240192}} | year=1968 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=9 | pages=417–450}} (Griess relates [p. 123] how Marshall Hall, as editor of The [[Journal of Algebra]], received a very short paper entitled "A simple group of order 604801." Yes, 604801 is prime.)
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| *{{Citation | last1=Janko | first1=Zvonimir | title=Symposia Mathematica (INDAM, Rome, 1967/68), Vol. 1 | publisher=[[Academic Press]] | location=Boston, MA | id={{MR|0244371}} | year=1969 | chapter=Some new simple groups of finite order. I | pages=25–64}}
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| * Wales, David B., "The uniqueness of the simple group of order 604800 as a subgroup of SL(6,4)", Journal of Algebra 11 (1969), 455 - 460.
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| * Wales, David B., "Generators of the Hall-Janko group as a subgroup of G2(4)", Journal of Algebra 13 (1969), 513–516, {{doi|10.1016/0021-8693(69)90113-6}}, MR0251133, ISSN 0021-8693
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| ==External links==
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| * [http://web.mat.bham.ac.uk/atlas/v2.0/spor/J2/ Atlas of Finite Group Representations: ''J''<sub>2</sub>]
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| {{DEFAULTSORT:Hall-Janko group}}
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| [[Category:Sporadic groups]]
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Yoshiko is her name but she doesn't like when individuals use her complete title. The favorite pastime for my children and me is taking part in crochet and now I'm trying to make cash with it. Arizona is her birth location and she will never move. Bookkeeping is what I do for a residing.
Also visit my homepage ... extended car warranty (please click chaddibaddi.com)