In the area of
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...
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 O'Nan group ''O'N'' or O'Nan–Sims group is a
sporadic simple group 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 d ...
: 2
93
457
3111931
: = 460815505920
: ≈ 5.
History
''O'Nan'' is one of the 26
sporadic groups and was found by in a study of
groups with a
Sylow 2-
subgroup of "
Alperin Alperin is a Jewish surname. Notable people with the surname include:
* J. L. Alperin
Jonathan Lazare Alperin (; born 1937) is an American mathematician specializing in the area of algebra known as group theory. He is notable for his work in gro ...
type", meaning
isomorphic to a Sylow 2-Subgroup of a group of type (Z/2
''n''Z ×Z/2
''n''Z ×Z/2
''n''Z).PSL
3(F
2). For the O'Nan group ''n'' = 2 and the
extension does not
split. The only other simple group with a Sylow 2-subgroup of Alperin type with ''n'' ≥ 2 is the
Higman–Sims group again with ''n'' = 2, but the extension splits.
The
Schur multiplier has order 3, and its
outer automorphism group has order 2. showed that O'Nan cannot be a
subquotient of the
monster group. Thus it is one of the 6 sporadic groups called the
pariahs.
Representations
showed that its triple cover has two 45-
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
al representations over the field with 7 elements, exchanged by an outer automorphism.
Maximal subgroups
and independently found the 13
conjugacy classes of
maximal subgroups of ''O'Nan'' as follows:
* L
3(7):2 (2 classes, fused by an
outer automorphism)
*
J1 The subgroup fixed by an outer
involution in ''O'Nan'':2.
* 4
2.L
3(4):2
1 The centralizer of an (inner)
involution in ''O'Nan''.
* (3
2:4 × A
6).2
* 3
4:2
1+4.D
10
* L
2(31) (2 classes, fused by an outer automorphism)
* 4
3.L
3(2)
*
M11 (2 classes, fused by an outer automorphism)
*
A7 (2 classes, fused by an outer automorphism)
O'Nan moonshine
In 2017 John F. R. Duncan, Michael H. Mertens, and Ken Ono proved theorems that establish an analogue of
monstrous moonshine for the O'Nan group. Their results "reveal a role for the O'Nan pariah group as a provider of hidden
symmetry to
quadratic forms and
elliptic curves." The O'Nan moonshine results "also represent the intersection of moonshine theory with the
Langlands program, which, since its inception in the 1960s, has become a driving force for research in
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
,
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and
mathematical physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
." .
An informal description of these developments was written by in ''
Quanta Magazine''.
Sources
*
*
*
*
*
*
*
External links
MathWorld: O'Nan Group*
{{DEFAULTSORT:O'Nan group
Sporadic groups