Janko group J4

In mathematics, the fourth Janko group "J"₄ is a sporadic finite simple group whose existence was suggested by Zvonimir Janko (1976), and then proven to uniquely exist by Simon Norton and others in 1980. It is the unique finite simple group of order $2^\{21\}cdot\; 3^3cdot\; 5cdot\; 7cdot\; 11^3cdot\; 23cdot\; 29cdot\; 31cdot\; 37cdot\; 43$. Janko found it by studying groups with an involution centralizer of the form 2¹⁺¹².3.(M₂₂:2). It has a modular representation of dimension 112 over the finite field of two 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. Ivanov (2004) has given a proof of existence and uniqueness that does not rely on computer calculations. It has a presentation in terms of three generators a, b, and c as:$a^2=b^3=c^2=(ab)^\{23\}=\; [a,b]\; ^\{12\}=\; [a,bab]\; ^5=\; [c,a]\; =$:$(ababab^\{-1\})^3(abab^\{-1\}ab^\{-1\})^3=(ab(abab^\{-1\})^3)^4=$:$[c,bab(ab^\{-1\})^2(ab)^3]\; =(bc^\{bab^\{-1\}abab^\{-1\}a\})^3=$:$((bababab)^3cc^\{(ab)^3b(ab)^6b\})^2=1.$

J₄ is one of the 6 sporadic simple groups called the pariahs, because they are not found within the Monster group. The order of the monster group is not divisible by 37 or 43.

J₄ has 13 conjugacy classes of maximal subgroups.
* 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 = F₈₁₂
* 43:14 = F₆₀₂
* 37:12 = F₄₄₄A Sylow 3-subgroup is a Heisenberg group: order 27, non-abelian, all non-trivial elements of order 3

References

*Ivanov, A. A. "The fourth Janko group." Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2004. xvi+233 pp. ISBN 0-19-852759-4 MathSciNet|id=2124803
*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.DOI|10.1016/0021-8693(76)90115-0 (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.
* [http://web.mat.bham.ac.uk/atlas/v2.0/spor/J4/ Atlas of Finite Group Representations: "J"₄]