View source for Janko group J3

{{Short description|Sporadic simple group}}
{{for|general background and history of the Janko sporadic groups|Janko group}}
{{DISPLAYTITLE:Janko group J<sub>3</sub>}}
{{Group theory sidebar |Finite}}

In the area of modern algebra known as [[group theory]], the '''Janko group''' ''J<sub>3</sub>'' or the '''Higman-Janko-McKay group''' ''HJM'' is a [[sporadic simple group]] of [[Order (group theory)|order]]
:&nbsp;&nbsp;&nbsp;50,232,960 = 2<sup>7</sup>{{·}}3<sup>5</sup>{{·}}5{{·}}17{{·}}19.

==History and properties==
''J<sub>3</sub>'' is one of the 26 [[Sporadic group]]s and was predicted by [[Zvonimir Janko]] in 1969 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 [[Janko group J2|''J<sub>2</sub>'']]). 
''J<sub>3</sub>'' was shown to exist by {{harvs|txt |author1-link=Graham Higman |first=Graham |last=Higman |author2-link=John McKay (mathematician) |first2=John |last2=McKay |year=1969}}.

In 1982 [[R. L. Griess]] showed that ''J<sub>3</sub>'' cannot be a [[subquotient]] of the [[monster group]].<ref>Griess (1982): p. 93: proof that  J<sub>3</sub> is a pariah.</ref>  Thus it is one of the 6 sporadic groups called the [[pariah group|pariahs]].

J<sub>3</sub> has an [[outer automorphism group]] of order 2 and a  [[Schur multiplier]] of order 3, and its triple cover has a unitary 9-dimensional [[Modular representation theory|representation]] over the [[finite field]] with 4 elements. {{harvtxt|Weiss|1982}} constructed it via an underlying geometry. It has a modular representation of dimension eighteen over the [[finite field]] with 9 elements.
It has a complex projective representation of dimension eighteen.

==Constructions==
=== Using matrices ===
J3 can be constructed by many different [[Generator (mathematics)|generators]].<ref>[http://brauer.maths.qmul.ac.uk/Atlas/spor/J3/ ATLAS page on J3]</ref> Two from the ATLAS list are 18x18 matrices over the [[finite field]] of order 9, with matrix multiplication carried out with [[finite field arithmetic]]:

<math>\left( \begin{matrix}
0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 \\
3 & 7 & 4 & 8 & 4 & 8 & 1 & 5 & 5 & 1 & 2 & 0 & 8 & 6 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 \\
4 & 8 & 6 & 2 & 4 & 8 & 0 & 4 & 0 & 8 & 4 & 5 & 0 & 8 & 1 & 1 & 8 & 5 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 \\
\end{matrix} \right)
</math>

and

<math>\left( \begin{matrix}
4 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
4 & 4 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 \\
2 & 7 & 4 & 5 & 7 & 4 & 8 & 5 & 6 & 7 & 2 & 2 & 8 & 8 & 0 & 0 & 5 & 0 \\
4 & 7 & 5 & 8 & 6 & 1 & 1 & 6 & 5 & 3 & 8 & 7 & 5 & 0 & 8 & 8 & 6 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
8 & 2 & 5 & 5 & 7 & 2 & 8 & 1 & 5 & 5 & 7 & 8 & 6 & 0 & 0 & 7 & 3 & 8 \\
\end{matrix} \right)
</math>

=== Using the subgroup PSL(2,16) ===
The automorphism group ''J''<sub>3</sub>:2 can be constructed by starting with the subgroup PSL(2,16):4 and adjoining 120 involutions, which are identified with the [[Sylow theorems|Sylow]] 17-subgroups. Note that these 120 involutions are outer elements of ''J''<sub>3</sub>:2. One then defines the following relation:

<math>\left(\begin{matrix}1&1\\1&0\end{matrix}\sigma t_{(\nu,\nu7)}\right)^5=1</math>

where <math>\sigma</math> is the [[Frobenius endomorphism|Frobenius automorphism]] or order 4, and <math>t_{(\nu,\nu7)}</math> is the unique 17-cycle that sends

<math>\infty\rightarrow0\rightarrow1\rightarrow7</math>

Curtis showed, using a computer, that this relation is sufficient to define ''J''<sub>3</sub>:2.<ref>{{citation| title=Symmetric Generationand existence of ''J''<sub>3</sub>:2, the automorphism group of the third Janko group| last1=Bradley | first1=J.D. | last2=Curtis | first2=R.T. | doi=10.1016/j.jalgebra.2005.09.046 |journal=Journal of Algebra | volume=304 | issue=1 | year=2006| pages=256–270 | doi-access=free }}</ref>

=== Using a presentation ===

In terms of generators a, b, c, and d its automorphism group J<sub>3</sub>:2 can be [[Presentation of a group|presented]] as
<math>a^{17} = b^8 = a^ba^{-2} = c^2 = b^cb^3 = (abc)^4 = (ac)^{17} = d^2 = [d, a] = [d, b] = (a^3b^{-3}cd)^5 = 1.</math>

A presentation for J<sub>3</sub> in terms of (different) generators a, b, c, d is
<math>a^{19} = b^9 = a^ba^2 = c^2 = d^2 = (bc)^2 = (bd)^2 = (ac)^3 = (ad)^3 = (a^2ca^{-3}d)^3 = 1.</math>

==Maximal subgroups==
{{harvtxt|Finkelstein|Rudvalis|1974}} found the 9 conjugacy classes of maximal subgroups of ''J<sub>3</sub>'' as follows:

{| class="wikitable"
|+ Maximal subgroups of ''J<sub>3</sub>''
|-
! No. !! Structure !! Order !! Index !! Comments
|-
|  1||L<sub>2</sub>(16):2                 ||align=right|8,160<br />=&nbsp;2<sup>5</sup>·3·5·17            ||align=right| 6,156<br />=&nbsp;2<sup>2</sup>·3<sup>4</sup>·19   ||   
|-
|2,3||L<sub>2</sub>(19)                   ||align=right|3,420<br />=&nbsp;2<sup>2</sup>·3<sup>2</sup>·5·19||align=right|14,688<br />=&nbsp;2<sup>5</sup>·3<sup>3</sup>·17   ||two classes, fused by an outer automorphism
|-
|  4||2<sup>4</sup>: (3 × A<sub>5</sub>)  ||align=right|2,880<br />=&nbsp;2<sup>6</sup>·3<sup>2</sup>·5   ||align=right|17,442<br />=&nbsp;2·3<sup>3</sup>·17·19            ||
|-
|  5||L<sub>2</sub>(17)                   ||align=right|2,448<br />=&nbsp;2<sup>4</sup>·3<sup>2</sup>·17  ||align=right|20,520<br />=&nbsp;2<sup>3</sup>·3<sup>3</sup>·5·19 ||centralizer of an outer automorphism of order 2
|-
|  6||(3 × A<sub>6</sub>):2<sub>2</sub>   ||align=right|2,160<br />=&nbsp;2<sup>4</sup>·3<sup>3</sup>·5   ||align=right|23,256<br />=&nbsp;2<sup>3</sup>·3<sup>2</sup>·17·19||normalizer of a subgroup of order 3 (class 3A)
|-
|  7||3<sup>2+1+2</sup>:8                 ||align=right|1,944<br />=&nbsp;2<sup>3</sup>·3<sup>5</sup>     ||align=right|25,840<br />=&nbsp;2<sup>4</sup>·5·17·19            ||normalizer of a Sylow 3-subgroup
|-
|  8||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|26,163<br />=&nbsp;3<sup>4</sup>·17·19              ||centralizer of involution
|-
|  9||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|43,605<br />=&nbsp;3<sup>3</sup>·5·17·19            ||
|}

== References ==
{{Reflist}}
*{{Citation | last1=Finkelstein | first1=L. | authorlink2=Arunas Rudvalis|last2=Rudvalis | first2=A. | title=The maximal subgroups of Janko's simple group of order 50,232,960 | doi=10.1016/0021-8693(74)90196-3 | year=1974 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=30 | issue=1–3 | pages=122–143 | mr=0354846| doi-access=free }}
* [[R. L. Griess]], Jr., ''The Friendly Giant'', Inventiones Mathematicae  69 (1982), 1-102. p.&nbsp;93: proof that J<sub>3</sub> is a pariah.
* {{Citation | last1=Higman | first1=Graham | author1-link=Graham Higman | last2=McKay | first2=John | author2-link=John McKay (mathematician) | title=On Janko's simple group of order 50,232,960 |mr=0246955 | year=1969 | journal=Bull. London Math. Soc. | volume=1 | pages=89–94; correction p. 219 | doi=10.1112/blms/1.1.89}}
*Z. Janko, ''Some new finite simple groups of finite order'', 1969  Symposia Mathematica (INDAM, Rome, 1967/68), Vol. 1  pp.&nbsp;25–64 Academic Press, London, and in ''The theory of finite groups'' (Edited by Brauer and Sah) p.&nbsp;63-64, Benjamin, 1969.{{MathSciNet|id=0244371}}
*{{cite journal |first=Richard |last=Weiss |title=A Geometric Construction of Janko's Group J<sub>3</sub> |journal=Mathematische Zeitschrift |issue=179 |pages=91–95 |year=1982|volume=179 |doi=10.1007/BF01173917 }}

==External links==
* [http://mathworld.wolfram.com/JankoGroups.html MathWorld: Janko Groups]
* [https://web.archive.org/web/20040706070806/http://web.mat.bham.ac.uk/atlas/v2.0/spor/J3/ Atlas of Finite Group Representations: ''J''<sub>3</sub>] version 2
* [http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/J3/ Atlas of Finite Group Representations: ''J''<sub>3</sub>] version 3

{{DEFAULTSORT:Janko group J3}}
[[Category:Sporadic groups]]