Cyclic (mathematics)

Last updated March 22, 2019

There are many terms in mathematics that begin with cyclic:

Cyclic chain rule, for derivatives, used in thermodynamics
Cyclic code, linear codes closed under cyclic permutations
Cyclic convolution, a method of combining periodic functions
Cycle decomposition (graph theory)
Cycle decomposition (group theory)
Cyclic extension, a field extension with cyclic Galois group
Graph theory:
- Cycle graph, a connected, 2-regular graph
- Cycle graph (algebra), a diagram representing the cycles determined by taking powers of group elements
- Circulant graph, a graph with cyclic symmetry
- Cycle (graph theory), a nontrivial path in some graph from a node to itself
- Cyclic graph, a graph containing at least one graph cycle
In group theory, a sub-field of abstract algebra, a group cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups.
In graph theory, a circulant graph is an undirected graph acted on by a cyclic group of symmetries which takes any vertex to any other vertex. It is sometimes called a cyclic graph, but this term also is given other meanings.
In graph theory, a cycle is a path of edges and vertices wherein a vertex is reachable from itself. There are several different types of cycles, principally a closed walk and a simple cycle; also, e.g., an element of the cycle space of the graph. If a graph contains no cycles it is referred to as being acyclic.
Cyclic group, a group generated by a single element
Cyclic homology, an approximation of K-theory used in non-commutative differential geometry
Cyclic module, a module generated by a single element
Cyclic notation, a way of writing permutations
Cyclic number, a number such that cyclic permutations of the digits are successive multiples of the number
Cyclic order, a ternary relation defining a way to arrange a set of objects in a circle
Cyclic permutation, a permutation with one nontrivial orbit
Cyclic polygon, a polygon which can be given a circumscribed circle
Cyclic shift, also known as circular shift
Cyclic symmetry, n-fold rotational symmetry of 3-dimensional space

In coding theory, a cyclic code is a block code, where the circular shifts of each codeword gives another word that belongs to the code. They are error-correcting codes that have algebraic properties that are convenient for efficient error detection and correction.

In graph theory, a cycle decomposition is a decomposition into cycles. Every vertex in a graph that has a cycle decomposition must have even degree.

In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices connected in a closed chain. The cycle graph with n vertices is called C_n. The number of vertices in C_n equals the number of edges, and every vertex has degree 2; that is, every vertex has exactly two edges incident with it.

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In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object.

In mathematics, a group is a set equipped with a binary operation which combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, and help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study.

In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object X is G = Sym(X).

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 S_n 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 S_n is n!.

In mathematics, permutation is the act of arranging the members of a set into a sequence or order, or, if the set is already ordered, rearranging (reordering) its elements—a process called permuting. Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set {1,2,3}, namely: (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). These are all the possible orderings of this three-element set. Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory.

In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of n elements is called the alternating group of degree n, or the alternating group on n letters and denoted by A_n or Alt(n).

In group theory, a branch of abstract algebra, a cyclic group or monogenous group is a group that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse. Each element can be written as a power of g in multiplicative notation, or as a multiple of g in additive notation. This element g is called a generator of the group.

In mathematics, and in particular the theory of group representations, the regular representation of a group G is the linear representation afforded by the group action of G on itself by translation.

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.

In mathematics, the concept of irreducibility is used in several ways.

In combinatorial mathematics a cycle index is a polynomial in several variables which is structured in such a way that information about how a group of permutations acts on a set can be simply read off from the coefficients and exponents. This compact way of storing information in an algebraic form is frequently used in combinatorial enumeration.

In mathematics, in the realm of group theory, a group is said to be thin if there is a finite upper bound on the girth of the Cayley graph induced by any finite generating set. The group is called fat if it is not thin.

In graph theory, the hypercube graph $Q n$ is the graph formed from the vertices and edges of an $n$ -dimensional hypercube. For instance, the cubical graph $Q 3$ is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube. $Q n$ has $2 n$ vertices, $2 n -1 n$ edges, and is a regular graph with $n$ edges touching each vertex.

In algebra, which is a broad division of mathematics, abstract algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. The term abstract algebra was coined in the early 20th century to distinguish this area of study from the other parts of algebra.

In graph theory, the modular decomposition is a decomposition of a graph into subsets of vertices called modules. A module is a generalization of a connected component of a graph. Unlike connected components, however, one module can be a proper subset of another. Modules therefore lead to a recursive (hierarchical) decomposition of the graph, instead of just a partition.

Cyclic (mathematics)

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