|
O'Nan Group
In the area of abstract algebra known as group theory, 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 and was found by in a study of groups with a Sylow 2-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 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-dimensional representations over the field with 7 elements, excha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the ''variety of groups''. History Before the nineteenth century, algebra meant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pariah Group
In group theory, the term pariah was introduced by Robert Griess in to refer to the six sporadic simple groups which are not subquotients 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 246320597611213317192329314147 .... The twenty groups which are subquotients, including the monster group itself, he dubbed the happy family. For example, the orders of ''J''4 and the Lyons Group ''Ly'' are divisible by 37. Since 37 does not divide the order of the monster, these cannot be subquotients of it; thus ''J''4 and ''Ly'' are pariahs. Three other sporadic groups were also shown to be pariahs by Griess in 1982, and the Janko Group J1 was shown to be the final pariah by Robert A. Wilson in 1986. The complete list is shown below. References * * Robert A. Wilson (1986)''Is J1 a subgroup of the m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. Widely seen as the single biggest project in modern mathematical research, the Langlands program has been described by Edward Frenkel as "a kind of grand unified theory of mathematics." The Langlands program consists of some very complicated theoretical abstractions, which can be difficult even for specialist mathematicians to grasp. To oversimplify, the fundamental lemma of the project posits a direct connection between the generalized fundamental representation of a finite field with its group extension to the automorphic forms under which it is invariant. This is accomplished through abstraction to higher dimensional integrati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions for: :y^2 = x^3 + ax + b for some coefficients and in . The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition , that is, being square-free in .) It is always understood that the curve is really sitting in the projective plane, with the point being the unique point at infinity. Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 to a fixed field , such as the real or complex numbers, and one speaks of a quadratic form over . If K=\mathbb R, and the quadratic form takes zero only when all variables are simultaneously zero, then it is a definite quadratic form, otherwise it is an isotropic quadratic form. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal group), differential geometry (Riemannian metric, second fundamental form), differential topology ( intersection forms of four-manifolds), and Lie theory (the Killing form). Quadratic forms are not to be confused with a quadratic equation, which has only one variable and includes terms of degree two or less. A quadratic form is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetry In Mathematics
Symmetry occurs not only in geometry, but also in other branches of mathematics. Symmetry is a type of invariance: the property that a mathematical object remains unchanged under a set of operations or transformations. Given a structured object ''X'' of any sort, a symmetry is a mapping of the object onto itself which preserves the structure. This can occur in many ways; for example, if ''X'' is a set with no additional structure, a symmetry is a bijective map from the set to itself, giving rise to permutation groups. If the object ''X'' is a set of points in the plane with its metric structure or any other metric space, a symmetry is a bijection of the set to itself which preserves the distance between each pair of points (i.e., an isometry). In general, every kind of structure in mathematics will have its own kind of symmetry, many of which are listed in the given points mentioned above. Symmetry in geometry The types of symmetry considered in basic geometry include reflec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. The monstrous moonshine is now known to be underlain by a vertex operator algebra called the moonshine module (or monster vertex algebra) constructed by Igor Frenkel, James Lepowsky, and Arne Meurman in 1988, which has the monster group as its group of symmetries. This vertex operator algebra is commonly interpreted as a structure underlying a two-dimensional conformal field theory, allowing physics to form a bridge between two mathematical areas. The conjectures made by Conway and Norton were proven by Richard Borcherds for the moonshine module in 1992 using the no-ghost theorem from string theory and the theory of vertex operator algebras and generalized Kac–Moody algebras. History In 1978, John McKay found that the first few ter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Group
In mathematics, an alternating group is the 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 by or Basic properties For , the group A''n'' is the commutator subgroup of the symmetric group S''n'' with index 2 and has therefore ''n''!/2 elements. It is the kernel of the signature group homomorphism explained under symmetric group. The group A''n'' is abelian if and only if and simple if and only if or . A5 is the smallest non-abelian simple group, having order 60, and the smallest non-solvable group. The group A4 has the Klein four-group V as a proper normal subgroup, namely the identity and the double transpositions , that is the kernel of the surjection of A4 onto . We have the exact sequence . In Galois theory, this map, or rather the corresponding map , corresponds to associating the Lagrange resolvent cubic to a quartic, which allow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathieu Group M11
In the area of modern algebra known as group theory, the Mathieu group ''M11'' is a sporadic simple group of order : 2432511 = 111098 = 7920. History and properties ''M11'' is one of the 26 sporadic groups and was introduced by . It is the smallest sporadic group and, along with the other four Mathieu groups, the first to be discovered. The Schur multiplier and the outer automorphism group are both trivial. ''M11'' is a sharply 4-transitive permutation group on 11 objects. It admits many generating sets of permutations, such as the pair (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) of permutations used by the GAP computer algebra system. Representations M11 has a sharply 4-transitive permutation representation on 11 points. The point stabilizer is sometimes denoted by M10, and is a non-split extension of the form A6.2 (an extension of the group of order 2 by the alternating group A6). This action is the automorphism group of a Steiner system S(4,5,11) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Involution (mathematics)
In mathematics, an involution, involutory function, or self-inverse function is a function that is its own inverse, : for all in the domain of . Equivalently, applying twice produces the original value. General properties Any involution is a bijection. The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (x \mapsto -x), reciprocation (x \mapsto 1/x), and complex conjugation (z \mapsto \bar z) in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the Beaufort polyalphabetic cipher. The composition of two involutions ''f'' and ''g'' is an involution if and only if they commute: . Involutions on finite sets The number of involutions, including the identity involution, on a set with elements is given by a recurrence relation found by Heinrich August Rothe in 1800: :a_0 = a_1 = 1 and a_n = a_ + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Janko Group J1
In the area of modern algebra known as group theory, the Janko group ''J1'' is a sporadic simple group of order : 233571119 = 175560 : ≈ 2. History ''J1'' is one of the 26 sporadic groups and was originally described by Zvonimir Janko in 1965. It is the only Janko group whose existence was proved by Janko himself and was the first sporadic group to be found since the discovery of the Mathieu groups in the 19th century. Its discovery launched the modern theory of sporadic groups. In 1986 Robert A. Wilson showed that ''J1'' cannot be a subgroup of the monster group. Thus it is one of the 6 sporadic groups called the pariahs. Properties The smallest faithful complex representation of ''J1'' has dimension 56.Jansen (2005), p.123 ''J1'' can be characterized abstractly as the unique simple group with abelian 2-Sylow subgroups and with an involution whose centralizer is isomorphic to the direct product of the group of order two and the alternating group A5 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 trivial center, then is said to be complete. An automorphism of a group which is not inner is called an outer automorphism. The cosets of with respect to outer automorphisms are then the elements of ; this is an instance of the fact that quotients of groups are not, in general, (isomorphic to) subgroups. If the inner automorphism group is trivial (when a group is abelian), the automorphism group and outer automorphism group are naturally identified; that is, the outer automorphism group does act on the group. For example, for the alternating group, , the outer automorphism group is usually the group of order 2, with exceptions noted below. Considering as a subgroup of the symmetric group, , conjugation by any odd permutation is an oute ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
