In mathematics, a permutation group is a group G whose elements are permutations of a given set M and whose group operation is the composition of permutations in G. The group of all permutations of a set M is the symmetric group of M, often written as Sym(M). The term permutation group thus means a subgroup of the symmetric group. If M = {1,2,...,n} then, Sym(M), the symmetric group on n letters is usually denoted by Sn.
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group Sn defined over a finite set of n symbols consists of the permutation operations that can be performed on the n symbols. Since there are n! possible permutation operations that can be performed on a tuple composed of n symbols, it follows that the number of elements of the symmetric group Sn is n!.
In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third one. It can be described as the symmetry group of a non-square rectangle (with the three non-identity elements being horizontal and vertical reflection and 180-degree rotation), as the group of bitwise exclusive or operations on two-bit binary values, or more abstractly as Z2 × Z2, the direct product of two copies of the cyclic group of order 2. It was named Vierergruppe (meaning four-group) by Felix Klein in 1884. It is also called the Klein group, and is often symbolized by the letter V or as K4.
In mathematics, Galois theory provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is in some sense simpler and better understood.
In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group acting on G. This can be understood as an example of the group action of G on the elements of G.
In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. Each such matrix, say P, represents a permutation of m elements and, when used to multiply another matrix, say A, results in permuting the rows or columns of the matrix A.
In mathematics, a generalized permutation matrix is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is exactly one nonzero entry in each row and each column. Unlike a permutation matrix, where the nonzero entry must be 1, in a generalized permutation matrix the nonzero entry can be any nonzero value. An example of a generalized permutation matrix is
In mathematics, a Young symmetrizer is an element of the group algebra of the symmetric group, constructed in such a way that, for the homomorphism from the group algebra to the endomorphisms of a vector space obtained from the action of on by permutation of indices, the image of the endomorphism determined by that element corresponds to an irreducible representation of the symmetric group over the complex numbers. A similar construction works over any field, and the resulting representations are called Specht modules. The Young symmetrizer is named after British mathematician Alfred Young.
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.
In mathematics, especially in an area of abstract algebra known as representation theory, a faithful representation ρ of a group on a vector space is a linear representation in which different elements of are represented by distinct linear mappings .
In mathematics, a permutation group G acting on a non-empty set X is called primitive if G acts transitively on X and G preserves no nontrivial partition of X, where nontrivial partition means a partition that isn't a partition into singleton sets or partition into one set X. Otherwise, if G is transitive and G does preserve a nontrivial partition, G is called imprimitive.
In mathematics, the expression Gelfand pair is a pair (G,K) consisting of a group G and a subgroup K that satisfies a certain property on restricted representations. The theory of Gelfand pairs is closely related to the topic of spherical functions in the classical theory of special functions, and to the theory of Riemannian symmetric spaces in differential geometry. Broadly speaking, the theory exists to abstract from these theories their content in terms of harmonic analysis and representation theory.
In mathematics, symmetrization is a process that converts any function in n variables to a symmetric function in n variables. Conversely, anti-symmetrization converts any function in n variables into an antisymmetric function.
In mathematics, the generalized symmetric group is the wreath product of the cyclic group of order m and the symmetric group of order n.
In mathematics, the Garnir relations give a way of expressing a basis of the Specht modules Vλ in terms of standard polytabloids.
In mathematical set theory, a permutation model is a model of set theory with atoms (ZFA) constructed using a group of permutations of the atoms. A symmetric model is similar except that it is a model of ZF and is constructed using a group of permutations of a forcing poset. One application is to show the independence of the axiom of choice from the other axioms of ZFA or ZF. Permutation models were introduced by Fraenkel (1922) and developed further by Mostowski (1938). Symmetric models were introduced by Paul Cohen.
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