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 of order : 2⁷3³5²7 = 604800 : ≈ 6.

History and properties

''J₂'' is one of the 26

Sporadic group In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. The ...

s and is also called Hall–Janko–Wales group. In 1969

Zvonimir Janko Zvonimir Janko (26 July 1932 – 12 April 2022) was a Croatian mathematician who was the eponym of the Janko groups, sporadic simple groups in group theory. The first few sporadic simple groups were discovered by Émile Léonard Mathieu, which ...

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 : 273551719 = 50232960. History and properties ''J3'' is one of the 26 Sp ...

). It was constructed by as a

rank 3 permutation group Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...

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 \oper ...

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 a t ...

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, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is 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 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, having order 246320597611213317192329314147 ...

; it is thus part of what

Robert Griess Robert Louis Griess, Jr. (born 1945, Savannah, Georgia) is a mathematician working on finite simple groups and vertex algebras. He is currently the John Griggs Thompson Distinguished University Professor of mathematics at University of Michigan. ...

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 : 221395472111323 : = 4157776806543360000 : ≈ 4. History and properties ''Co1'' is one of the 26 sporadic grou ...

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

Representations

It is a subgroup of

index Index (or its plural form 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 a Halo megastru ...

two of the group of automorphisms of the

Hall–Janko graph In the mathematical field of graph theory, the Hall–Janko graph, also known as the Hall-Janko-Wales graph, is a 36- regular undirected graph with 100 vertices and 1800 edges. It is a rank 3 strongly regular graph with parameters (100,36,14,12) ...

, 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 In mathematics, Hurwitz's automorphisms theorem bounds the order of the group of automorphisms, via orientation-preserving conformal mappings, of a compact Riemann surface of genus ''g'' > 1, stating that the number of such automorphisms ...

, 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. This importance stems from its connection to Hurwitz surfaces, namely Riemann surfaces of genus ''g'' with the largest possible order, ...

. 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 In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other wor ...

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. * (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. ''Journal of Algebra'' was founded by Graham Higman, who was its editor from 1964 to 1 ...

, 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)", 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, , ,

External links

MathWorld: Janko Groups

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

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