Hall–Janko group

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In the area of modern algebra known as group theory, the Janko group J₂ or the Hall-Janko group HJ is a sporadic simple group of order

   2⁷ · 3³ · 5² · 7 = 604800

≈ 6×10⁵.

History and properties

J₂ is one of the 26 Sporadic groups and is also called Hall–Janko–Wales group. In 1969 Zvonimir Janko predicted J₂ as one of two new simple groups having 2¹⁺⁴:A₅ as a centralizer of an involution (the other is the Janko group J3). It was constructed by Marshall Hall and David Wales (1968) 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 J₂ 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.A₁₀₀. The double cover 2.J₂ occurs as a subgroup of the Conway group Co₀.

J₂ 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.

Representations

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 is also a subgroup of index two of the group of automorphisms of the Hall–Janko Near Octagon,^[1] 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 w² + w + 1 = 0, then J₂ is generated by the two matrices

${\displaystyle {\mathbf {A} }={\begin{pmatrix}w^{2}&w^{2}&0&0&0&0\\1&w^{2}&0&0&0&0\\1&1&w^{2}&w^{2}&0&0\\w&1&1&w^{2}&0&0\\0&w^{2}&w^{2}&w^{2}&0&w\\w^{2}&1&w^{2}&0&w^{2}&0\end{pmatrix}$

and

${\displaystyle {\mathbf {B} }={\begin{pmatrix}w&1&w^{2}&1&w^{2}&w^{2}\\w&1&w&1&1&w\\w&w&w^{2}&w^{2}&1&0\\0&0&0&0&1&1\\w^{2}&1&w^{2}&w^{2}&w&w^{2}\\w^{2}&1&w^{2}&w&w^{2}&w\end{pmatrix}.}$

These matrices satisfy the equations

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

(Note that matrix multiplication on a finite field of order 4 is defined slightly differently from ordinary matrix multiplication. See Finite field § Field with four elements for the specific addition and multiplication tables, with w the same as a and w² the same as 1 + a.)

J₂ 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 G₂(4). There is only one conjugacy class of J₂ in G₂(4). Every subgroup J₂ contained in G₂(4) extends to a subgroup J₂:2 = Aut(J₂) in G₂(4):2 = Aut(G₂(4)) (G₂(4) extended by the field automorphisms of F₄). G₂(4) is in turn isomorphic to a subgroup of the Conway group Co₁.

Maximal subgroups

There are 9 conjugacy classes of maximal subgroups of J₂. 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

Conjugacy classes

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

References

• Robert L. Griess, Jr., "Twelve Sporadic Groups", Springer-Verlag, 1998.
• Hall, Marshall; Wales, David (1968), "The simple group of order 604,800", Journal of Algebra, 9 (4): 417–450, doi:10.1016/0021-8693(68)90014-8, ISSN 0021-8693, MR 0240192 (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.)
• Janko, Zvonimir (1969), "Some new simple groups of finite order. I", Symposia Mathematica (INDAM, Rome, 1967/68), Vol. 1, Boston, MA: Academic Press, pp. 25–64, MR 0244371
• Wales, David B., "The uniqueness of the simple group of order 604800 as a subgroup of SL(6,4)", Journal of Algebra 11 (1969), 455–460.
• Wales, David B., "Generators of the Hall–Janko group as a subgroup of G2(4)", Journal of Algebra 13 (1969), 513–516, doi:10.1016/0021-8693(69)90113-6, MR0251133, ISSN 0021-8693

External links

• MathWorld: Janko Groups
• Atlas of Finite Group Representations: J₂
• The subgroup lattice of J₂