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In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have
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) is usually denoted by Sn, and may be called the symmetric group on n letters.
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 defined over a finite set of symbols consists of the permutations that can be performed on the symbols. Since there are such permutation operations, the order of the symmetric group is .
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set.
In mathematics, when X is a finite set with at least two elements, the permutations of X fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity of a permutation of X can be defined as the parity of the number of inversions for σ, i.e., of pairs of elements x, y of X such that x < y and σ(x) > σ(y).
In mathematics, anticommutativity is a specific property of some non-commutative operations. In mathematical physics, where symmetry is of central importance, these operations are mostly called antisymmetric operations, and are extended in an associative setting to cover more than two arguments. Swapping the position of two arguments of an antisymmetric operation yields a result which is the inverse of the result with unswapped arguments. The notion inverse refers to a group structure on the operation's codomain, possibly with another operation, such as addition.
In mathematics, and in particular in group theory, a cyclic permutation is a permutation of the elements of some set X which maps the elements of some subset S of X to each other in a cyclic fashion, while fixing all other elements of X. If S has k elements, the cycle is called a k-cycle. Cycles are often denoted by the list of their elements enclosed with parentheses, in the order to which they are permuted.
In mathematics, the braid group on n strands, also known as the Artin braid group, is the group whose elements are equivalence classes of n-braids, and whose group operation is composition of braids. Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids ; in mathematical physics where Artin's canonical presentation of the braid group corresponds to the Yang–Baxter equation ; and in monodromy invariants of algebraic geometry.
In mathematics, the outer automorphism group of a group, G, is the quotient, Aut(G) / Inn(G), where Aut(G) is the automorphism group of G and Inn(G) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted Out(G). If Out(G) is trivial and G has a trivial center, then G is said to be complete.
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.
In mathematics, a permutation group G acting on a non-empty finite set X is called primitive if G acts transitively on X and the only partitions the G-action preserves are the trivial partitions into either a single set or into |X| singleton sets. Otherwise, if G is transitive and G does preserve a nontrivial partition, G is called imprimitive.
In mathematics, a 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, a Specht module is one of the representations of symmetric groups studied by Wilhelm Specht (1935). They are indexed by partitions, and in characteristic 0 the Specht modules of partitions of n form a complete set of irreducible representations of the symmetric group on n points.
In mathematics, the immanant of a matrix was defined by Dudley E. Littlewood and Archibald Read Richardson as a generalisation of the concepts of determinant and permanent.
In mathematics, more specifically in multilinear algebra, an alternating multilinear map is a multilinear map with all arguments belonging to the same vector space that is zero whenever any pair of arguments is equal. More generally, the vector space may be a module over a commutative ring.
In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments:
In homotopy theory, a branch of mathematics, the Barratt–Priddy theorem expresses a connection between the homology of the symmetric groups and mapping spaces of spheres. The theorem is also often stated as a relation between the sphere spectrum and the classifying spaces of the symmetric groups via Quillen's plus construction.
In algebra, Solomon's descent algebra of a Coxeter group is a subalgebra of the integral group ring of the Coxeter group, introduced by Solomon (1976).
In mathematics, there are two natural interpretations of the place-permutation action of symmetric groups, in which the group elements act on positions or places. Each may be regarded as either a left or a right action, depending on the order in which one chooses to compose permutations. There are just two interpretations of the meaning of "acting by a permutation " but these lead to four variations, depending whether maps are written on the left or right of their arguments. The presence of so many variations often leads to confusion. When regarding the group algebra of a symmetric group as a diagram algebra it is natural to write maps on the right so as to compute compositions of diagrams from left to right.
Algebraic Eraser (AE) is an anonymous key agreement protocol that allows two parties, each having an AE public–private key pair, to establish a shared secret over an insecure channel. This shared secret may be directly used as a key, or to derive another key that can then be used to encrypt subsequent communications using a symmetric key cipher. Algebraic Eraser was developed by Iris Anshel, Michael Anshel, Dorian Goldfeld and Stephane Lemieux. SecureRF owns patents covering the protocol and unsuccessfully attempted to standardize the protocol as part of ISO/IEC 29167-20, a standard for securing radio-frequency identification devices and wireless sensor networks.
References
- Nyman, Kathryn L. (2003), "The peak algebra of the symmetric group", J. Algebraic Combin., 17 (3): 309–322, doi: 10.1023/A:1025000905826 , MR 2001673