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 Harada–Norton group ''HN'' is a
sporadic simple 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 ...
of
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
: 2
143
65
671119
: = 273030912000000
: ≈ 3.
History and properties
''HN'' is one of the 26 sporadic groups and was found by and ).
Its
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 \op ...
is trivial and its
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 ...
has order 2.
''HN'' has an involution whose
centralizer
In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', ...
is of the form 2.HS.2, where HS is the
Higman-Sims group (which is how Harada found it).
The prime 5 plays a special role in the group. For example, it centralizes an element of order 5 in 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
2463205976112133171923293141475 ...
(which is how Norton found it), and as a result acts naturally on a
vertex operator algebra
In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications, vertex operator algebras have proven useful ...
over the field with 5 elements . This implies that it acts on a 133 dimensional algebra over F
5 with a commutative but nonassociative product, analogous to the
Griess algebra In mathematics, the Griess algebra is a commutative non-associative algebra on a real vector space of dimension 196884 that has the Monster group ''M'' as its automorphism group. It is named after mathematician R. L. Griess, who constructed it in 1 ...
.
Generalized monstrous moonshine
Conway and Norton suggested in their 1979 paper that
monstrous moonshine
In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular, the ''j'' function. The term was coined by John Conway and Simon P. Norton in 1979 ...
is not limited to the monster, but that similar phenomena may be found for other groups. Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups.
To recall, the prime number 5 plays a special role in the group and for ''HN'', the relevant McKay-Thompson series is
where one can set the constant term (),
:
and ''η''(''τ'') is the
Dedekind eta function
In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string ...
.
Maximal subgroups
found the 14 conjugacy classes of
maximal subgroup
In mathematics, the term maximal subgroup is used to mean slightly different things in different areas of algebra.
In group theory, a maximal subgroup ''H'' of a group ''G'' is a proper subgroup, such that no proper subgroup ''K'' contains ''H' ...
s of ''HN'' as follows:
* A
12
* 2.HS.2
* U
3(8):3
* 2
1+8.(A
5 × A
5).2
* (D
10 × U
3(5)).2
* 5
1+4.2
1+4.5.4
* 2
6.U
4(2)
* (A
6 × A
6).D
8
* 2
3+2+6.(3 × L
3(2))
* 5
2+1+2.4.A
5
* M
12:2 (Two classes, fused by an outer automorphism)
* 3
4:2.(A
4 × A
4).4
* 3
1+4:4.A
5
References
*
*
*S. P. Norton, ''F and other simple groups'', PhD Thesis, Cambridge 1975.
*
*
External links
MathWorld: Harada–Norton GroupAtlas of Finite Group Representations: Harada–Norton group
{{DEFAULTSORT:Harada-Norton group
Sporadic groups