Hall–Janko group

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In mathematics the Hall-Janko group HJ, of order 604800, is also called the second Janko group J₂, 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.

We may also express it in terms of a modular representation of dimension six over the field of four elements; if in characteristic two we have w² + w + 1 = 0, then J₂ is generated by the two matrices

and

These matrices satisfy the equations

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.

[edit] 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 MR0244371
• Atlas of Finite Group Representations: J₂