View source for Janko group J2

{{Short description|Sporadic simple group}}
{{for|general background and history of the Janko sporadic groups|Janko group}}
{{DISPLAYTITLE:Janko group J<sub>2</sub>}}
{{Group theory sidebar |Finite}}

In the area of modern algebra known as [[group theory]], the '''Janko group''' ''J<sub>2</sub>'' or the '''Hall-Janko group''' ''HJ''  is a [[sporadic simple group]] of [[Order (group theory)|order]]
:&nbsp;&nbsp;&nbsp;604,800 = 2<sup>7</sup>{{·}}3<sup>3</sup>{{·}}5<sup>2</sup>{{·}}7
: ≈ 6{{e|5}}.

==History and properties==
''J<sub>2</sub>'' is one of the 26 [[Sporadic group]]s and is also called '''Hall–Janko–Wales group'''.  In 1969 [[Zvonimir Janko]] predicted J<sub>2</sub> as one of two new simple groups having 2<sup>1+4</sup>:A<sub>5</sub> as a centralizer of an involution (the other is the [[Janko group J3]]). It was constructed by {{harvs|txt |authorlink=Marshall Hall (mathematician) |first1=Marshall|last1=Hall |first2=David|last2=Wales |year=1968}} as a [[rank 3 permutation group]] on 100 points.

Both the [[Schur multiplier]] and the [[outer automorphism group]] have order 2. As a permutation group on 100 points J<sub>2</sub> has [[involution (mathematics)#group theory|involutions]] moving all 100 points and involutions moving just 80 points. The former involutions are 
products of 25 double transportions, an odd number, and hence lift to 4-elements in the [[Covering_groups_of_the_alternating_and_symmetric_groups|double cover]] 2.A<sub>100</sub>. The double cover 2.J<sub>2</sub> occurs as a [[Conway_group#Suzuki_chain_of_product_groups|subgroup]] of the Conway group Co<sub>0</sub>.

J<sub>2</sub> is the only one of the 4 Janko groups that is a [[subquotient]] 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 Co1]], it is therefore part of the second generation of the Happy Family.

== Representations ==
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. It is also a subgroup of index two of the group of automorphisms of the Hall–Janko [[Near Polygon|Near Octagon]],<ref>{{Cite web|url=http://www.win.tue.nl/~aeb/graphs/HJ315.html|title = The near octagon on 315 points}}</ref> leading to a permutation representation of degree 315.

It has a [[modular representation]] of dimension six over the field of four elements; if in [[characteristic (algebra)|characteristic]] two we have {{nowrap|''w''<sup>2</sup> + ''w'' + 1 {{=}} 0}}, then J<sub>2</sub> is generated by the two matrices

:<math>
{\mathbf A} = \begin{pmatrix}
w^2 & w^2 &   0 &   0 &   0 & 0 \\ 
  1 & w^2 &   0 &   0 &   0 & 0 \\ 
  1 &   1 & w^2 & w^2 &   0 & 0 \\ 
  w &   1 &   1 & w^2 &   0 & 0 \\ 
  0 & w^2 & w^2 & w^2 &   0 & w \\ 
w^2 &   1 & w^2 &   0 & w^2 & 0
\end{pmatrix}
</math>

and

:<math>
{\mathbf B} = \begin{pmatrix}
  w & 1 & w^2 &   1 & w^2 & w^2 \\ 
  w & 1 &   w &   1 &   1 & w \\ 
  w & w & w^2 & w^2 &   1 & 0 \\ 
  0 & 0 &   0 &   0 &   1 & 1 \\ 
w^2 & 1 & w^2 & w^2 &   w & w^2 \\ 
w^2 & 1 & w^2 &   w & w^2 & w
\end{pmatrix}.
</math>

These matrices satisfy the equations

:<math>
{\mathbf A}^2 = {\mathbf B}^3 = ({\mathbf A}{\mathbf B})^7 = 
({\mathbf A}{\mathbf B}{\mathbf A}{\mathbf B}{\mathbf B})^{12} = 1.
</math>

(Note that matrix multiplication on a finite field of order 4 is defined slightly differently from ordinary matrix multiplication. See {{section link|Finite field|Field with four elements}} for the specific addition and multiplication tables, with ''w'' the same as ''a'' and ''w''{{sup|2}} the same as ''1 + a''.)

J<sub>2</sub> is thus a [[Hurwitz group]], a finite homomorphic image of the [[(2,3,7) triangle group]].

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 is only one conjugacy class of J<sub>2</sub> in ''G''<sub>2</sub>(4). Every subgroup J<sub>2</sub> contained in ''G''<sub>2</sub>(4) extends to a subgroup J<sub>2</sub>:2{{nbsp}}= Aut(J<sub>2</sub>) in ''G''<sub>2</sub>(4):2{{nbsp}}= Aut(''G''<sub>2</sub>(4)) (''G''<sub>2</sub>(4) extended by the field automorphisms of '''F'''<sub>4</sub>). ''G''<sub>2</sub>(4) is in turn isomorphic to a subgroup of the [[Conway group]] Co<sub>1</sub>.

==Maximal subgroups==
There are 9 [[conjugacy class|conjugacy classes]] of [[maximal subgroup|maximal subgroups]] of ''J<sub>2</sub>''. Some are here described in terms of action on the Hall–Janko graph.

{| class="wikitable"
|+ Maximal subgroups of ''J<sub>2</sub>''
|-
! No. !! Structure !! Order !! Index !! Comments
|-
|1||U<sub>3</sub>(3)                   ||align=right|6,048<br />=&nbsp;2<sup>5</sup>·3<sup>3</sup>·7||align=right|   100<br />=&nbsp;2<sup>2</sup>·5<sup>2</sup>    ||one-point stabilizer, with orbits of sizes 1, 36, and 63; 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.
|-
|2||3.PGL<sub>2</sub>(9)               ||align=right|2,160<br />=&nbsp;2<sup>4</sup>·3<sup>3</sup>·5||align=right|   280<br />=&nbsp;2<sup>3</sup>·5·7              ||normalizer of a subgroup of order 3 (class 3A); has a subquotient A<sub>6</sub>
|-
|3||{{nowrap|2{{su|a=l|b=&nbsp;&ndash;|p=1+4}}}}:A<sub>5</sub>      ||align=right|1,920<br />=&nbsp;2<sup>7</sup>·3·5            ||align=right|   315<br />=&nbsp;3<sup>2</sup>·5·7              ||centralizer of involution of class 2A moving 80 points
|-
|4||2<sup>2+4</sup>:(3 × S<sub>3</sub>)||align=right|1,152<br />=&nbsp;2<sup>7</sup>·3<sup>2</sup>  ||align=right|   525<br />=&nbsp;3·5<sup>2</sup>·7              ||
|-
|5||A<sub>4</sub> × A<sub>5</sub>      ||align=right|  720<br />=&nbsp;2<sup>4</sup>·3<sup>2</sup>·5||align=right|   840<br />=&nbsp;2<sup>3</sup>·3·5·7            ||contains 2<sup>2</sup> × A<sub>5</sub> (of order 240), centralizer of 3 involutions each moving 100 points
|-
|6||A<sub>5</sub> × D<sub>10</sub>     ||align=right|  600<br />=&nbsp;2<sup>3</sup>·3·5<sup>2</sup>||align=right| 1,008<br />=&nbsp;2<sup>4</sup>·3<sup>2</sup>·7  ||normalizer of a subgroup of order 5 (classes 5A, 5B)
|-
|7||PGL<sub>2</sub>(7)                 ||align=right|  336<br />=&nbsp;2<sup>4</sup>·3·7            ||align=right| 1,800<br />=&nbsp;2<sup>3</sup>·3<sup>2</sup>·5<sup>2</sup>||
|-
|8||5<sup>2</sup>:D<sub>12</sub>       ||align=right|  300<br />=&nbsp;2<sup>2</sup>·3·5<sup>2</sup>||align=right| 2,016<br />=&nbsp;2<sup>5</sup>·3<sup>2</sup>·7  ||
|-
|9||A<sub>5</sub>                      ||align=right|   60<br />=&nbsp;2<sup>2</sup>·3·5            ||align=right|10,080<br />=&nbsp;2<sup>5</sup>·3<sup>2</sup>·5·7||
|}                                                     

==Conjugacy classes==
The maximum order of any element is 15. As permutations, elements act on the 100 vertices of the Hall–Janko graph.
{| class="wikitable" style="margin: 1em auto;"
|-
! Order || No. elements || Cycle structure and conjugacy
|-
| 1 = 1 || 1 = 1 || 1 class
|-
| rowspan="2" | 2 = 2 || 315 = 3<sup>2</sup> · 5 · 7 || 2<sup>40</sup>, 1 class
|-
| <!-- 2 = 2 ||--> 2520 = 2<sup>3</sup> · 3<sup>2</sup> · 5 · 7 || 2<sup>50</sup>, 1 class
|-
| rowspan="2" | 3 = 3 || 560 = 2<sup>4</sup> · 5 · 7 || 3<sup>30</sup>, 1 class
|-
| <!-- 3 = 3 ||--> 16800 = 2<sup>5</sup> · 3 · 5<sup>2</sup> · 7 || 3<sup>32</sup>, 1 class
|-
| 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
|-
| rowspan="2" | 5 = 5 || 4032 = 2<sup>6</sup> · 3<sup>2</sup> · 7 || 5<sup>20</sup>, 2 classes, power equivalent
|-
| <!-- 5 = 5 ||--> 24192 = 2<sup>7</sup> · 3<sup>3</sup> · 7 || 5<sup>20</sup>, 2 classes, power equivalent
|-
| 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
|-
| <!-- 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
|-
| 7 = 7 || 86400 = 2<sup>7</sup> · 3<sup>3</sup> · 5<sup>2</sup> || 7<sup>14</sup>, 1 class
|-
| 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
|-
| rowspan="2" | 10 = 2 · 5 || 60480 = 2<sup>6</sup> · 3<sup>3</sup> · 5 · 7 || 10<sup>10</sup>, 2 classes, power equivalent
|-
| <!-- 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
|-
| 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
|-
| 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
|}

== References ==
{{Reflist}}
* [[Robert L. Griess]], Jr., "Twelve Sporadic Groups", Springer-Verlag, 1998.
*{{Citation | last1=Hall | first1=Marshall | author-link=Marshall Hall (mathematician)|last2=Wales | first2=David | title=The simple group of order 604,800 | doi=10.1016/0021-8693(68)90014-8 |mr=0240192 | year=1968 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=9 | issue=4 | pages=417–450| doi-access=free }} (Griess relates [p.&nbsp;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.)
*{{Citation | last1=Janko | first1=Zvonimir | title=Symposia Mathematica (INDAM, Rome, 1967/68), Vol. 1 | publisher=[[Academic Press]] | location=Boston, MA |mr=0244371 | year=1969 | chapter=Some new simple groups of finite order. I | pages=25–64}}
* 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.
* 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}}, {{MR|0251133}}, {{issn|0021-8693}}

==External links==
* [http://mathworld.wolfram.com/JankoGroups.html MathWorld: Janko Groups]
* [https://web.archive.org/web/20160303165718/http://web.mat.bham.ac.uk/atlas/v2.0/spor/J2/ Atlas of Finite Group Representations: ''J''<sub>2</sub>]
* [https://web.archive.org/web/20140908022328/http://homepages.ulb.ac.be/~tconnor/atlaslat/j2.pdf The subgroup lattice of ''J''<sub>2</sub>]
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