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 ''J
2'' or the Hall-Janko group ''HJ'' is a
sporadic simple group of
order
: 2
73
35
27 = 604800
: ≈ 6.
History and properties
''J
2'' 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 J
2 as one of two new simple groups having 2
1+4:A
5 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 J
2 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
100. The double cover 2.J
2 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 Co
0.
J
2 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
24632059761121331719232931414759 ...
; 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 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 J
2 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
2 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.
The matrix representation given above constitutes an embedding into
Dickson's group
''G''2(4). There is only one conjugacy class of J
2 in ''G''
2(4). Every subgroup J
2 contained in ''G''
2(4) extends to a subgroup J
2:2= Aut(J
2) in ''G''
2(4):2= Aut(''G''
2(4)) (''G''
2(4) extended by the field automorphisms of F
4). ''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 autom ...
Co
1.
Maximal subgroups
There are 9
conjugacy classes of
maximal subgroups of ''J
2''. Some are here described in terms of action on the Hall–Janko graph.
* U
3(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.S
4, which contains 6 additional involutions.
* 3.PGL(2,9) order 2160 – has a subquotient A
6
* 2
1+4:A
5 order 1920 – centralizer of involution moving 80 points
* 2
2+4:(3 × S
3) order 1152
* A
4 × A
5 order 720
:Containing 2
2 × A
5 (order 240), centralizer of 3 involutions each moving 100 points
* A
5 × D
10 order 600
* PGL(2,7) order 336
* 5
2:D
12 order 300
* A
5 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 ...
, 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 GroupsAtlas of Finite Group Representations: ''J''2The subgroup lattice of ''J''2
{{DEFAULTSORT:Janko group J2
Sporadic groups