- Hall–Janko group
In
mathematics , the Hall-Janko group "HJ", is a finite simplesporadic 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 theHall-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:mathbf A} = left ( egin{matrix}w^2 & w^2 & 0 & 0 & 0 & 0 \ 1 & w^2 & 0 & 0 & 0 & 0 \ 1 & 1 & w^2 & w^2 & 0 & 0 \ w & 1 & 1 & w^2 & 0 & 0 \ 0 & w^2 & w^2 & w^2 & 0 & w \ w^2 & 1 & w^2 & 0 & w^2 & 0 end{matrix} ight )
and
:mathbf B} = left ( egin{matrix}w & 1 & w^2 & 1 & w^2 & w^2 \ w & 1 & w & 1 & 1 & w \ w & w & w^2 & w^2 & 1 & 0 \ 0 & 0 & 0 & 0 & 1 & 1 \ w^2 & 1 & w^2 & w^2 & w & w^2 \ w^2 & 1 & w^2 & w & w^2 & w end{matrix} ight )
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 group s that is a section of theMonster group ; it is thus part of whatRobert Griess calls the Happy Family. It is also found in theConway 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 subgroup s. 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]
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