|
Automorphisms Of The Symmetric And Alternating Groups
In group theory, a branch of mathematics, the automorphisms and outer automorphisms of the symmetric groups and alternating groups are both standard examples of these automorphisms, and objects of study in their own right, particularly the exceptional outer automorphism of S6, the symmetric group on 6 elements. Summary Generic case * n\neq 2,6: \operatorname(\mathrm_n) = \mathrm_n, and thus \operatorname(\mathrm_n) = \mathrm_1. :Formally, \mathrm_n is complete and the natural map \mathrm_n \to \operatorname(\mathrm_n) is an isomorphism. * n\neq 1,2,6: \operatorname(\mathrm_n)=\mathrm_n/\mathrm_n=\mathrm_2, and the outer automorphism is conjugation by an odd permutation. * n\neq 2,3,6: \operatorname(\mathrm_n)=\operatorname(\mathrm_n)=\mathrm_n :Indeed, the natural maps \mathrm_n \to \operatorname(\mathrm_n) \to \operatorname(\mathrm_n) are isomorphisms. Exceptional cases * n=1,2: trivial: :: \operatorname(\mathrm_1)=\operatorname(\mathrm_1)=\operatorname(\mathrm_1)=\operat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 (mathematics), fields, and vector spaces, can all be seen as groups endowed with additional operation (mathematics), operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and Standard Model, three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exotic Map
Exotic may refer to: Mathematics and physics * Exotic R4, a differentiable 4-manifold, homeomorphic but not diffeomorphic to the Euclidean space R4 *Exotic sphere, a differentiable ''n''-manifold, homeomorphic but not diffeomorphic to the ordinary ''n''-sphere *Exotic atom, an atom with one or more electrons replaced by other negatively charged particles *Exotic hadron **Exotic baryon, bound states of 3 quarks and additional particles **Exotic meson, non-quark model mesons *Exotic matter, a hypothetical concept of particle physics Music * "Exotic" (1963 song), a song by The Sentinals from the 1963 album ''Surf Crazy - Original Surfin' Hits'' * "Exotic" (Lil Baby song), 2018 * "Exotic" (Priyanka Chopra song), a 2012 song by Priyanka Chopra featuring Pitbull Flora and fauna *Exotic pet *Exotic Shorthair, a breed of cat *Exotic species (or introduced species), a species not native to an area Other *Exotic dancer, a type of dancer or stripper *Exotic derivative, a type of financia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, an affine transformation is an automorphism of an affine space (Euclidean spaces are specific affine spaces), that is, a function which maps an affine space onto itself while preserving both the dimension of any affine subspaces (meaning that it sends points to points, lines to lines, planes to planes, and so on) and the ratios of the lengths of parallel line segments. Consequently, sets of parallel affine subspaces remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. If is the point set of an affine space, then every affine transformation on can be repre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Group
In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a point. They are named after F. G. Frobenius. Structure Suppose ''G'' is a Frobenius group consisting of permutations of a set ''X''. A subgroup ''H'' of ''G'' fixing a point of ''X'' is called a Frobenius complement. The identity element together with all elements not in any conjugate of ''H'' form a normal subgroup called the Frobenius kernel ''K''. (This is a theorem due to ; there is still no proof of this theorem that does not use character theory, although see .) The Frobenius group ''G'' is the semidirect product of ''K'' and ''H'': :G=K\rtimes H. Both the Frobenius kernel and the Frobenius complement have very restricted structures. proved that the Frobenius kernel ''K'' is a nilpotent group. If ''H'' has even order then ''K'' is abelian. The Frobenius complement ''H'' has the property t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean-Pierre Serre
Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the inaugural Abel Prize in 2003. Biography Personal life Born in Bages, Pyrénées-Orientales, France, to pharmacist parents, Serre was educated at the Lycée de Nîmes and then from 1945 to 1948 at the École Normale Supérieure in Paris. He was awarded his doctorate from the Sorbonne in 1951. From 1948 to 1954 he held positions at the Centre National de la Recherche Scientifique in Paris. In 1956 he was elected professor at the Collège de France, a position he held until his retirement in 1994. His wife, Professor Josiane Heulot-Serre, was a chemist; she also was the director of the Ecole Normale Supérieure de Jeunes Filles. Their daughter is the former French diplomat, historian and writer Claudine Monteil. The French mathematician Denis S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transitive Group Action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group ''acts'' on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. For example, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron. A group action on a vector space is called a representation of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of , the group of the invertible matrices of dimension over a field . The symmetric group acts on any set with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Faithful Group Action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group ''acts'' on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. For example, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron. A group action on a vector space is called a representation of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of , the group of the invertible matrices of dimension over a field . The symmetric group acts on any set with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Line
In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; for example, two distinct projective lines in a projective plane meet in exactly one point (there is no "parallel" case). There are many equivalent ways to formally define a projective line; one of the most common is to define a projective line over a field ''K'', commonly denoted P1(''K''), as the set of one-dimensional subspaces of a two-dimensional ''K''-vector space. This definition is a special instance of the general definition of a projective space. The projective line over the reals is a manifold; see real projective line for details. Homogeneous coordinates An arbitrary point in the projective line P1(''K'') may be represented by an equivalence class of ''homogeneous coordinates'', which take the form of a pair : _1 : x_2/mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod when is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number and every positive integer there are fields of order p^k, all of which are isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set which is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Linear Group
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associated projective space P(''V''). Explicitly, the projective linear group is the quotient group :PGL(''V'') = GL(''V'')/Z(''V'') where GL(''V'') is the general linear group of ''V'' and Z(''V'') is the subgroup of all nonzero scalar transformations of ''V''; these are quotiented out because they act trivially on the projective space and they form the kernel of the action, and the notation "Z" reflects that the scalar transformations form the center of the general linear group. The projective special linear group, PSL, is defined analogously, as the induced action of the special linear group on the associated projective space. Explicitly: :PSL(''V'') = SL(''V'')/SZ(''V'') where SL(''V'') is the special linear group over ''V'' and SZ(''V'') is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sylow Subgroup
In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the classification of finite simple groups. For a prime number p, a Sylow ''p''-subgroup (sometimes ''p''-Sylow subgroup) of a group G is a maximal p-subgroup of G, i.e., a subgroup of G that is a ''p''-group (meaning its cardinality is a power of p, or equivalently, the order of every group element is a power of p) that is not a proper subgroup of any other p-subgroup of G. The set of all Sylow p-subgroups for a given prime p is sometimes written \text_p(G). The Sylow theorems assert a partial converse to Lagrange's theorem. Lagrange's theorem states that for any finite group G the order ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Factorization
In graph theory, a factor of a graph ''G'' is a spanning subgraph, i.e., a subgraph that has the same vertex set as ''G''. A ''k''-factor of a graph is a spanning ''k''- regular subgraph, and a ''k''-factorization partitions the edges of the graph into disjoint ''k''-factors. A graph ''G'' is said to be ''k''-factorable if it admits a ''k''-factorization. In particular, a 1-factor is a perfect matching, and a 1-factorization of a ''k''-regular graph is an edge coloring with ''k'' colors. A 2-factor is a collection of cycles that spans all vertices of the graph. 1-factorization If a graph is 1-factorable (ie, has a 1-factorization), then it has to be a regular graph. However, not all regular graphs are 1-factorable. A ''k''-regular graph is 1-factorable if it has chromatic index ''k''; examples of such graphs include: * Any regular bipartite graph. Hall's marriage theorem can be used to show that a ''k''-regular bipartite graph contains a perfect matching. One can then remove t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
