Hall–Janko group

In mathematics, the Hall-Janko group "HJ", is a finite simple sporadic group of order 604800. It is also called the second Janko group "J"₂, or the Hall-Janko-Wales group, since it was predicted by Janko and constructed by Hall and Wales. It is a subgroup of index two of the group of automorphisms of the Hall-Janko graph, leading to a permutation representation of degree 100.

It has a modular representation of dimension six over the field of four elements; if in characteristic two we have"w"² + "w" + 1 = 0, then J₂ is generated by the two matrices

:

and

:

These matrices satisfy the equations

:$\{mathbf\; A\}^2\; =\; \{mathbf\; B\}^3\; =\; (\{mathbf\; A\}\{mathbf\; B\})^7\; =\; (\{mathbf\; A\}\{mathbf\; B\}\{mathbf\; A\}\{mathbf\; B\}\{mathbf\; B\})^\{12\}\; =\; 1.$

J₂ is thus a Hurwitz group, a finite homomorphic image of the (2,3,7) triangle group.

J₂ is the only one of the 4 Janko groups that is a section of the Monster group; it is thus part of what Robert Griess calls the Happy Family. It is also found in the Conway group Co₁, and is therefore part of the second generation of the Happy Family.

Griess relates [p. 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.

J₂ has 9 conjugacy classes of maximal subgroups. Some are here described in terms of action on the Hall-Janko graph.

* U₃(3) order 6048 - one-point stabilizer, with orbits of 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.S₄, which contains 6 additional involutions.

* 3.PGL(2,9) order 2160 - has a subquotient A₆

* 2¹⁺⁴:A₅ order 1920 - centralizer of involution moving 80 points

* 2²⁺⁴:(3 × S₃) order 1152

* A₄ × A₅ order 720

:Containing 2² × A₅ (order 240), centralizer of 3 involutions each moving 100 points

* A₅ × D₁₀ order 600

* PGL(2,7) order 336

* 5²:D₁₂ order 300

* A₅ order 60

Janko predicted both J₂ and J₃ as simple groups having 2¹⁺⁴:A₅ as a centralizer of an involution.

Number of elements of each order

The maximum order of any element is 15. As permutations, elements act on the 100 vertices of the Hall-Janko graph.

Order	No. elements	Cycle structure and conjugacy
1 = 1	1 = 1	1 class
2 = 2	315 = 32 · 5 · 7	240, 1 class
	2520 = 23 · 32 · 5 · 7	250, 1 class
3 = 3	560 = 24 · 5 · 7	330, 1 class
	16800 = 25 · 3 · 52 · 7	332, 1 class
4 = 22	6300 = 22 · 32 · 52 · 7	26420, 1 class
5 = 5	4032 = 26 · 32 · 7	520, 2 classes, power equivalent
	24192 = 27 · 33 · 7	520, 2 classes, power equivalent
6 = 2 · 3	25200 = 24 · 32 · 52 · 7	2436612, 1 class
	50400 = 25 · 32 · 52 · 7	22616, 1 class
7 = 7	86400 = 27 · 33 · 52	714, 1 class
8 = 23	75600 = 24 · 33 · 52 · 7	2343810, 1 class
10 = 2 · 5	60480 = 26 · 33 · 5 · 7	1010, 2 classes, power equivalent
	120960 = 27 · 33 · 5 · 7	54108, 2 classes, power equivalent
12 = 22 · 3	50400 = 25 · 32 · 52 · 7	324262126, 1 class
15 = 3 · 5	80640 = 28 · 32 · 5 · 7	52156, 2 classes, power equivalent

References

* Robert L. Griess, Jr., "Twelve Sporadic Groups", Springer-Verlag, 1998.
* Marshall Hall, Jr. and David Wales, "The Simple Group of Order 604,800", Journal of Algebra, 9 (1968), 417-450.
* Z. Janko, "Some new finite simple groups of finite order", 1969 Symposia Mathematica (INDAM, Rome, 1967/68), Vol. 1 pp. 25-64 Academic Press, London MathSciNet|id=0244371
* [http://web.mat.bham.ac.uk/atlas/v2.0/spor/J2/ Atlas of Finite Group Representations: "J"₂]