In the area of
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
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
: 460,815,505,920 = 2
93
457
3111931 ≈ 5.
History
''O'N'' is one of the 26
sporadic groups and was found by in a study of
groups with a
Sylow 2-
subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation ...
of "
Alperin type", meaning
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
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). The following simple groups have Sylow 2-subgroups of Alperin type:
* For the
Chevalley group ''G''
2(q), if q is congruent to 3 or 5 mod 8, ''n = 1'' and the extension does not
split.
* For the
Steinberg group 3''D''
4(q), if q is congruent to 3 or 5 mod 8, ''n = 1'' and the extension does not split.
* For the
alternating group
In mathematics, an alternating group is the Group (mathematics), group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted ...
A
8, ''n = 1'' and the extension splits.
* For the O'Nan group, ''n'' = 2 and the extension does not split.
* For the
Higman-Sims group, ''n'' = 2 and the extension splits.
The
Schur multiplier has order 3, and its
outer automorphism group has order 2. showed that ''O'N'' 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 coo ...
al representations over the field with 7 elements, exchanged by an outer automorphism.
The degrees of
irreducible representations of the O'Nan group are 1, 10944, 13376, 13376, 25916, ... .
Maximal subgroups
and independently found the 13
conjugacy class
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 ...
es of
maximal subgroups of ''O'N'' as follows:
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
Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
to
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong t ...
s and
elliptic curves." The O'Nan moonshine results "also represent the intersection of moonshine theory with the
Langlands program
In mathematics, the Langlands program is a set of conjectures about connections between number theory, the theory of automorphic forms, and geometry. It was proposed by . It seeks to relate the structure of Galois groups in algebraic number t ...
, which, since its inception in the 1960s, has become a driving force for research in
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
,
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
and
mathematical physics
Mathematical physics is 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 the de ...
." .
An informal description of these developments was written by in ''
Quanta Magazine''.
Sources
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External links
MathWorld: O'Nan Group*
{{DEFAULTSORT:O'Nan group
Sporadic groups