Janko Group J4

In the area of modern algebra known as group theory, the Janko group ''J₄'' is a sporadic simple group of order
:   2²¹3³5711³2329313743
: = 86775571046077562880
: ≈ 9.

History

''J₄'' is one of the 26 Sporadic groups. Zvonimir Janko found J₄ in 1975 by studying groups with an involution centralizer of the form 2^{1 + 12}.3.(M₂₂:2). 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 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. and gave computer-free proofs of uniqueness. and gave a computer-free proof of existence by constructing it as an amalgams of groups 2¹⁰:SL₅(2) and (2¹⁰:2⁴:A₈):2 over a group 2¹⁰:2⁴:A₈.

The Schur multiplier and the outer automorphism group are both trivial.

Since 37 and 43 are not supersingular primes, ''J₄'' cannot be a subquotient of the monster group. Thus it is one of the 6 sporadic groups called the 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 2¹¹M₂₄. 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
:\begin
a^2 &=b^3=c^2=(ab)^= ,b= ,bab5= ,a \left ((ab)^2ab^ \right)^3 \left (ab(ab^)^2 \right)^3=\left (ab \left (abab^ \right )^3 \right )^4 \\
&=\left ,(ba)^2 b^ab^ (ab)^3 \right \left (bc^ \right )^3= \left ((bababab)^3 c c^ \right )^2=1.
\end

Maximal subgroups

found the 13 conjugacy classes of maximal subgroups of ''J₄'' as follows:

* 2¹¹:M₂₄ - containing Sylow 2-subgroups and Sylow 3-subgroups; also containing 2¹¹:(M₂₂:2), centralizer of involution of class 2B
* 2¹⁺¹².3.(M₂₂:2) - centralizer of involution of class 2A - containing Sylow 2-subgroups and Sylow 3-subgroups
* 2¹⁰:PSL(5,2)
* 2³⁺¹².(S₅ × PSL(3,2)) - containing Sylow 2-subgroups
* U₃(11):2
* M₂₂:2
* 11¹⁺²:(5 × GL(2,3)) - normalizer of Sylow 11-subgroup
* PSL(2,32):5
* PGL(2,23)
* U₃(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

*
*D.J. Benson ''The simple group J₄'', 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
*
*
*Ivanov, A. A. ''The fourth Janko group.'' Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2004. xvi+233 pp.
*Z. Janko, ''A new finite simple group of order 86,775,570,046,077,562,880 which possesses M₂₄ and the full covering group of M₂₂ as subgroups'', J. Algebra 42 (1976) 564-596. (The title of this paper is incorrect, as the full covering group of M₂₂ was later discovered to be larger: center of order 12, not 6.)
*
*S. P. Norton ''The construction of J₄'' in ''The Santa Cruz conference on finite groups'' (Ed. Cooperstein, Mason) Amer. Math. Soc 1980.

External links

MathWorld: Janko Groups

Atlas of Finite Group Representations: ''J''₄
version 2
Atlas of Finite Group Representations: ''J''₄
version 3

{{DEFAULTSORT:Janko group J3
Sporadic groups