View source for Janko group J4

{{DISPLAYTITLE:Janko group J<sub>4</sub>}}
{{Group theory sidebar |Finite}}

In the area of modern algebra known as [[Group theory|group theory]], the '''Janko group''' ''J<sub>4</sub>'' is a sporadic simple group of [[Order (group theory)|order]]
:&nbsp;&nbsp;&nbsp;2<sup>21</sup>{{·}}3<sup>3</sup>{{·}}5{{·}}7{{·}}11<sup>3</sup>{{·}}23{{·}}29{{·}}31{{·}}37{{·}}43
: = 86775571046077562880
: ≈ 9{{e|19}}.

==History==
''J<sub>4</sub>'' is one of the 26 [[Sporadic group]]s.  [[Biography:Zvonimir Janko|Zvonimir Janko]] found  J<sub>4</sub> in 1975 by studying groups with an involution centralizer of the form 2<sup>1 + 12</sup>.3.(M<sub>22</sub>:2).  Its existence and uniqueness was shown using computer calculations by [[Biography:Simon P. Norton|Simon P. Norton]] and others in 1980.   It has a modular representation of dimension 112 over the [[Finite field|finite field]] with 2 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</sub>.

The [[Schur multiplier]] and the [[Outer automorphism group|outer automorphism group]] are both [[Trivial group|trivial]].

Since  37 and 43 are not [[Supersingular prime (moonshine theory)|supersingular]] primes, ''J<sub>4</sub>'' cannot be a [[Subquotient|subquotient]] of the [[Monster group|monster group]]. Thus it is one of the 6 sporadic groups called the [[Pariah group|pariahs]].

==Representations==

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.

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.

==Presentation==

It has a presentation in terms of three generators a, b, and c as
:<math>\begin{align}
a^2 &=b^3=c^2=(ab)^{23}=[a,b]^{12}=[a,bab]^5=[c,a]= \left ((ab)^2ab^{-1} \right)^3 \left (ab(ab^{-1})^2 \right)^3=\left (ab \left (abab^{-1} \right )^3 \right )^4 \\
&=\left [c,(ba)^2 b^{-1}ab^{-1} (ab)^3 \right]= \left (bc^{(bab^{-1}a)^2} \right )^3= \left ((bababab)^3 c c^{(ab)^3b(ab)^6b} \right )^2=1.
\end{align}</math>

==Maximal subgroups==
{{harvtxt|Kleidman|Wilson|1988}} found the 13 conjugacy classes of maximal subgroups of ''J<sub>4</sub>'' as follows:

* 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
* 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
* 2<sup>10</sup>:PSL(5,2)
* 2<sup>3+12</sup>.(S<sub>5</sub> × PSL(3,2)) - containing Sylow 2-subgroups
* U<sub>3</sub>(11):2
* M<sub>22</sub>:2
* 11<sup>1+2</sup>:(5 × GL(2,3)) - normalizer of Sylow 11-subgroup
* PSL(2,32):5
* PGL(2,23)
* U<sub>3</sub>(3) - containing Sylow 3-subgroups
* 29:28 Frobenius group
* 43:14 Frobenius group
* 37:12 Frobenius group
A Sylow 3-subgroup is a [[Heisenberg group]]: order 27, non-abelian, all non-trivial elements of order 3.

== References ==

*{{Citation | last1=Aschbacher | first1=Michael |  last2=Segev | first2=Yoav | title=The uniqueness of groups of type J<sub>4</sub> | doi=10.1007/BF01232280 |mr=1117152 | year=1991 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=105 | issue=3 | pages=589–607| s2cid=121529060 }}
*D.J. Benson ''The simple group J<sub>4</sub>'', PhD Thesis, Cambridge 1981,  https://web.archive.org/web/20110610013308/http://www.maths.abdn.ac.uk/~bensondj/papers/b/benson/the-simple-group-J4.pdf
*{{Citation | last1=Ivanov | first1=A. A. | title=A presentation for J<sub>4</sub> | doi=10.1112/plms/s3-64.2.369 |mr=1143229 | year=1992 | journal=Proceedings of the London Mathematical Society |series=Third Series | issn=0024-6115 | volume=64 | issue=2 | pages=369–396}}
*{{Citation | last1=Ivanov | first1=A. A. | last2=Meierfrankenfeld | first2=Ulrich | title=A computer-free construction of J<sub>4</sub> | doi=10.1006/jabr.1999.7851 |mr=1707666 | year=1999 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=219 | issue=1 | pages=113–172| doi-access=free }}
* {{cite book|last=Ivanov|first=A. A.|title=The Fourth Janko Group|series=Oxford Mathematical Monographs|year=2004|publisher=Clarendon Press|location=Oxford|isbn=0-19-852759-4}}{{MR|2124803}}
*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.)
*{{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 |series=Third Series | issn=0024-6115 | volume=56 | issue=3 | pages=484–510}}
*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.

==External links==
* [http://mathworld.wolfram.com/JankoGroups.html MathWorld: Janko Groups]
* [https://web.archive.org/web/20040706071034/http://web.mat.bham.ac.uk/atlas/v2.0/spor/J4/ Atlas of Finite Group Representations: ''J''<sub>4</sub>] version 2
*[http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/J4/ Atlas of Finite Group Representations: ''J''<sub>4</sub>] version 3

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

{{Sourceattribution|Janko group J4}}