Harada-Norton group
Encyclopedia
In the mathematical field of group theory
, the Harada–Norton group HN (found by and ) is a sporadic
simple group
of order
Its Schur multiplier
is trivial and its outer automorphism group
has order 2.
The Harada–Norton group has an involution whose centralizer 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
(which is how Norton found it), and as a result acts naturally on a vertex operator algebra
over the field with 5 elements .
This implies that it acts on a 133 dimensional algebra over F5 with a commutative but nonassociative product, analogous to the Griess algebra
.
described the maximal subgroups.
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, the Harada–Norton group HN (found by and ) is a sporadic
Sporadic group
In the mathematical field of group theory, a sporadic group is one of the 26 exceptional groups in the classification of finite simple groups. A simple group is a group G that does not have any normal subgroups except for the subgroup consisting only of the identity element, and G itself...
simple group
Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
of order
Order (group theory)
In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
- 214 · 36 · 56 · 7 · 11 · 19
- = 273030912000000
- ≈ 3 · 1014.
Its Schur multiplier
Schur multiplier
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2 of a group G.It was introduced by in his work on projective representations.-Examples and properties:...
is trivial and its outer automorphism group
Outer automorphism group
In mathematics, the outer automorphism group of a group Gis the quotient Aut / Inn, where Aut is the automorphism group of G and Inn is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted Out...
has order 2.
The Harada–Norton group has an involution whose centralizer is of the form 2.HS.2, where HS is the Higman-Sims group
Higman-Sims group
In the mathematical field of group theory, the Higman–Sims group HS is a sporadic simple group found by of orderThe Higman–Sims group was discovered in 1967, when Higman and Sims were attending a presentation by Marshall Hall on the Hall–Janko 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
Monster group
In the mathematical field of group theory, the Monster group M or F1 is a group of finite order:...
(which is how Norton found it), and as a result acts naturally on a vertex operator algebra
Vertex operator algebra
In mathematics, a vertex operator algebra is an algebraic structure that plays an important role in conformal field theory and related areas of physics...
over the field with 5 elements .
This implies that it acts on a 133 dimensional algebra over F5 with a commutative but nonassociative product, analogous to the Griess algebra
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 1980 and subsequently used it in 1982 to construct M...
.
described the maximal subgroups.