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 group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...

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 : 2⁹3⁴57³111931 : = 460815505920 : ≈ 5.

History

''O'Nan'' is one of the 26

sporadic groups 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. Th ...

and was found by in a study of

groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

with a Sylow 2-

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 " Alperin 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).PSL₃(F₂). For the O'Nan group ''n'' = 2 and the

extension Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * E ...

does not

split Split(s) or The Split may refer to: Places * Split, Croatia, the largest coastal city in Croatia * Split Island, Canada, an island in the Hudson Bay * Split Island, Falkland Islands * Split Island, Fiji, better known as Hạfliua Arts, entertai ...

. The only other simple group with a Sylow 2-subgroup of Alperin type with ''n'' ≥ 2 is the

Higman–Sims group In the area of modern algebra known as group theory, the Higman–Sims group HS is a sporadic simple group of order : 29⋅32⋅53⋅7⋅11 = 44352000 : ≈ 4. The Schur multiplier has order 2, the outer automorphism ...

again with ''n'' = 2, but the extension splits. 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 ...

has order 3, 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 t ...

has order 2. showed that O'Nan cannot be 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 ...

. Thus it is one of the 6 sporadic groups called the

pariahs Pariah may refer to: * A member of the Paraiyar caste in the Indian state of Tamil Nadu * Pariah state, a country whose behavior does not conform to norms * Outcast (person) Science and mathematics * Pariah dog, a type of semi-feral dog * ''Pa ...

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 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 wo ...

es 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 ''O'Nan'' as follows: * L₃(7):2 (2 classes, fused by an

outer automorphism 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 ...

) * J₁ The subgroup fixed by an outer

involution Involution may refer to: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...

in ''O'Nan'':2. * 4₂.L₃(4):2₁ The centralizer of an (inner)

in ''O'Nan''. * (3²:4 × A₆).2 * 3⁴:2¹⁺⁴.D₁₀ * L₂(31) (2 classes, fused by an outer automorphism) * 4³.L₃(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 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. ...

for the O'Nan group. Their results "reveal a role for the O'Nan pariah group as a provider of hidden

symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...

to quadratic forms and

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...

s." The O'Nan moonshine results "also represent the intersection of moonshine theory with the

Langlands program In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic num ...

, 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 arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

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