In the area of modern algebra known as

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...

, the Janko group ''J₂'' or the Hall-Janko group ''HJ'' is a

sporadic simple group In the mathematical classification of finite simple groups, there are a number of groups which do not fit into any infinite family. These are called the sporadic simple groups, or the sporadic finite groups, or just the sporadic groups. A simpl ...

order Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood ...

: 604,800 = 2⁷3³5²7 : ≈ 6.

History and properties

''J₂'' is one of the 26

Sporadic group In the mathematical classification of finite simple groups, there are a number of groups which do not fit into any infinite family. These are called the sporadic simple groups, or the sporadic finite groups, or just the sporadic groups. A simpl ...

s 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 In the area of modern algebra known as group theory, the Janko group ''J3'' or the Higman-Janko-McKay group ''HJM'' is a sporadic simple group of order : 50,232,960 = 273551719. History and properties ''J3'' is one of the 26 ...

). It was constructed by as a

rank 3 permutation group In mathematical finite group theory, a rank 3 permutation group acts transitively on a set such that the stabilizer of a point has 3 orbits. The study of these groups was started by . Several of the sporadic simple groups were discovered as rank 3 ...

on 100 points. Both the

Schur multiplier In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations. Examples and properties The Schur multiplier \ope ...

and the

outer automorphism group In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has ...

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 In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ...

of the Conway group Co₀. J₂ is the only one of the 4 Janko groups that is a

subquotient In the mathematical fields of category theory and abstract algebra, a subquotient is a quotient object of a subobject. Subquotients are particularly important in abelian categories, and in group theory, where they are also known as sections, thou ...

of the

monster group In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group; it has order : : = 2463205976112133171923293 ...

; it is thus part of what Robert Griess calls the Happy Family. Since it is also found in the

Conway group Co1 In the area of modern algebra known as group theory, the Conway group ''Co1'' is a sporadic simple group of order : 4,157,776,806,543,360,000 : = 221395472111323 : ≈ 4. History and properties ''Co1'' is one of the 26 sporad ...

, it is therefore part of the second generation of the Happy Family.

Representations

It is a subgroup of

index Index (: indexes or indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on the Halo Array in the ...

two of the group of automorphisms of the Hall–Janko graph, leading to a

permutation representation In mathematics, the term permutation representation of a (typically finite) group G can refer to either of two closely related notions: a representation of G as a group of permutations, or as a group of permutation matrices. The term also refers ...

of degree 100. It is also a subgroup of index two of the group of automorphisms of the Hall–Janko Near Octagon, leading to a permutation representation of degree 315. It has a

modular representation Modular representation theory is a branch of mathematics, and is the part of representation theory that studies linear representations of finite groups over a field ''K'' of positive characteristic ''p'', necessarily a prime number. As well as h ...

of dimension six over the field of four elements; if in characteristic two we have , then J₂ is generated by the two matrices :

= \begin
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

and :

= \begin
  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.

These matrices satisfy the equations :

^2 = ^3 = ()^7 = 
()^ = 1.

(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''.) J₂ is thus a Hurwitz group, a finite homomorphic image of the

(2,3,7) triangle group In the theory of Riemann surfaces and hyperbolic geometry, the triangle group (2,3,7) is particularly important for its connection to Hurwitz surfaces, namely Riemann surfaces of genus ''g'' with the largest possible order, 84(''g'' − 1), of it ...

. The matrix representation given above constitutes an embedding into Dickson's group ''G''₂(4). There is only one conjugacy class of J₂ in ''G''₂(4). Every subgroup J₂ contained in ''G''₂(4) extends to a subgroup J₂:2= Aut(J₂) in ''G''₂(4):2= Aut(''G''₂(4)) (''G''₂(4) extended by the field automorphisms of F₄). ''G''₂(4) is in turn isomorphic to a subgroup of the

Conway group In the area of modern algebra known as group theory, the Conway groups are the three sporadic simple groups Co1, Co2 and Co3 along with the related finite group Co0 introduced by . The largest of the Conway groups, Co0, is the group of auto ...

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.

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. * (Griess relates . 123how Marshall Hall, as editor of The

Journal of Algebra ''Journal of Algebra'' (ISSN 0021-8693) is an international mathematical research journal in algebra. An imprint of Academic Press, it is published by Elsevier Elsevier ( ) is a Dutch academic publishing company specializing in scientific, te ...

, received a very short paper entitled "A simple group of order 604801." Yes, 604801 is prime.) * * Wales, David B., "The uniqueness of the simple group of order 604800 as a subgroup of SL(6,4)",

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, , ,

External links

MathWorld: Janko Groups

Atlas of Finite Group Representations: ''J''₂

The subgroup lattice of ''J''₂
{{DEFAULTSORT:Janko group J2 Sporadic groups