In the area of modern algebra known as group theory, the Janko group J<sub>2</sub> or the Hall-Janko group HJ is a sporadic simple group of order
J<sub>2</sub> is one of the 26 Sporadic groups and is also called HallâÂÂJankoâÂÂWales group. In 1969 Zvonimir Janko predicted J<sub>2</sub> as one of two new simple groups having 2<sup>1+4</sup>:A<sub>5</sub> as a centralizer of an involution (the other is the Janko group J3). It was constructed by as a rank 3 permutation group on 100 points.
Both the Schur multiplier and the outer automorphism group have order 2. As a permutation group on 100 points J<sub>2</sub> has involutions moving all 100 points and involutions moving just 80 points. The former involutions are products of 25 double transportions, an odd number, and hence lift to 4-elements in the double cover 2.A<sub>100</sub>. The double cover 2.J<sub>2</sub> occurs as a subgroup of the Conway group Co<sub>0</sub>.
J<sub>2</sub> is the only one of the 4 Janko groups that is a subquotient of the monster group; it is thus part of what Robert Griess calls the Happy Family. Since it is also found in the Conway group Co1, it is therefore part of the second generation of the Happy Family.
It is a subgroup of index two of the group of automorphisms of the HallâÂÂJanko graph, leading to a permutation representation of degree 100. It is also a subgroup of index two of the group of automorphisms of the HallâÂÂJanko Near Octagon, leading to a permutation representation of degree 315.
It has a modular representation of dimension six over the field of four elements; if in characteristic two we have , then J<sub>2</sub> is generated by the two matrices
and
These matrices satisfy the equations
(Note that matrix multiplication on a finite field of order 4 is defined slightly differently from ordinary matrix multiplication. See for the specific addition and multiplication tables, with w the same as a and w the same as 1 + a.)
J<sub>2</sub> is thus a Hurwitz group, a finite homomorphic image of the (2,3,7) triangle group.
The matrix representation given above constitutes an embedding into Dickson's group G<sub>2</sub>(4). There is only one conjugacy class of J<sub>2</sub> in G<sub>2</sub>(4). Every subgroup J<sub>2</sub> contained in G<sub>2</sub>(4) extends to a subgroup J<sub>2</sub>:2= Aut(J<sub>2</sub>) in G<sub>2</sub>(4):2= Aut(G<sub>2</sub>(4)) (G<sub>2</sub>(4) extended by the field automorphisms of F<sub>4</sub>). G<sub>2</sub>(4) is in turn isomorphic to a subgroup of the Conway group Co<sub>1</sub>.
There are 9 conjugacy classes of maximal subgroups of J<sub>2</sub>. Some are here described in terms of action on the HallâÂÂJanko graph.
The maximum order of any element is 15. As permutations, elements act on the 100 vertices of the HallâÂÂJanko graph.