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# Cyclic group

In algebra, a cyclic group or monogenous group is a group that is generated by a single element. [1]

106 relations: Abelian extension, Abelian group, Additive group, American Mathematical Monthly, Associative property, Automorphism, Cayley graph, Celtic knot, Character theory, Circle, Circle group, Circulant graph, Commutative property, Commutative ring, Complex number, Composite number, Coprime integers, Coset, Countable set, Cycle graph, Cycle graph (algebra), Cyclic module, Cyclic number (group theory), Cyclic order, Cyclically ordered group, Dicyclic group, Direct product, Direct product of groups, Distributive lattice, Divisibility rule, Divisor, Duality (order theory), End (graph theory), Endomorphism ring, Euler's totient function, Exponentiation, Field (mathematics), Field extension, Finite field, Finite group, Finitely generated abelian group, Finitely generated group, Flag of Hong Kong, Flag of the Isle of Man, Frieze group, Frobenius endomorphism, Galois group, Generating set of a group, Graduate Studies in Mathematics, Graph automorphism, ... Expand index (56 more) »

## Abelian extension

In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian.

## Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

An additive group is a group of which the group operation is to be thought of as addition in some sense.

## American Mathematical Monthly

The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894.

## Associative property

In mathematics, the associative property is a property of some binary operations.

## Automorphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself.

## Cayley graph

In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a graph that encodes the abstract structure of a group.

## Celtic knot

Celtic knots, called Icovellavna, (snaidhm Cheilteach, cwlwm Celtaidd) are a variety of knots and stylized graphical representations of knots used for decoration, used extensively in the Celtic style of Insular art.

## Character theory

In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix.

## Circle

A circle is a simple closed shape.

## Circle group

In mathematics, the circle group, denoted by T, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers The circle group forms a subgroup of C&times;, the multiplicative group of all nonzero complex numbers.

## Circulant graph

In graph theory, a circulant graph is an undirected graph that has a cyclic group of symmetries which takes any vertex to any other vertex.

## Commutative property

In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.

## Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative.

## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

## Composite number

A composite number is a positive integer that can be formed by multiplying together two smaller positive integers.

## Coprime integers

In number theory, two integers and are said to be relatively prime, mutually prime, or coprime (also written co-prime) if the only positive integer (factor) that divides both of them is 1.

## Coset

In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, then Only when H is normal will the set of right cosets and the set of left cosets of H coincide, which is one definition of normality of a subgroup.

## Countable set

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.

## Cycle graph

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.

## Cycle graph (algebra)

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.

## Cyclic module

In mathematics, more specifically in ring theory, a cyclic module is a module that is generated by one element over a ring.

## Cyclic number (group theory)

A cyclic number is a natural number n such that n and φ(n) are coprime.

## Cyclic order

In mathematics, a cyclic order is a way to arrange a set of objects in a circle.

## Cyclically ordered group

In mathematics, a cyclically ordered group is a set with both a group structure and a cyclic order, such that left and right multiplication both preserve the cyclic order.

## Dicyclic group

In group theory, a dicyclic group (notation Dicn or Q4n) is a member of a class of non-abelian groups of order 4n (n > 1).

## Direct product

In mathematics, one can often define a direct product of objects already known, giving a new one.

## Direct product of groups

In group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted.

## Distributive lattice

In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other.

## Divisibility rule

A divisibility rule is a shorthand way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits.

## Divisor

In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder.

## Duality (order theory)

In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop or Pd.

## End (graph theory)

In the mathematics of infinite graphs, an end of a graph represents, intuitively, a direction in which the graph extends to infinity.

## Endomorphism ring

In abstract algebra, the endomorphism ring of an abelian group X, denoted by End(X), is the set of all endomorphisms of X (i.e., the set of all homomorphisms of X into itself) endowed with an addition operation defined by pointwise addition of functions and a multiplication operation defined by function composition.

## Euler's totient function

In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to.

## Exponentiation

Exponentiation is a mathematical operation, written as, involving two numbers, the base and the exponent.

## Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

## Field extension

In mathematics, and in particular, algebra, a field E is an extension field of a field F if E contains F and the operations of F are those of E restricted to F. Equivalently, F is a subfield of E. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers.

## 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.

## Finite group

In abstract algebra, a finite group is a mathematical group with a finite number of elements.

## Finitely generated abelian group

In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements x1,..., xs in G such that every x in G can be written in the form with integers n1,..., ns.

## Finitely generated group

In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination (under the group operation) of finitely many elements of the finite set S and of inverses of such elements.

## Flag of Hong Kong

The flag of Hong Kong features a white, stylised, five-petal Hong Kong orchid tree (''Bauhinia blakeana'') flower in the centre of a red field.

## Flag of the Isle of Man

The flag of the Isle of Man or flag of Mann (brattagh Vannin) is a triskelion, composed of three armoured legs with golden spurs, upon a red background.

## Frieze group

In mathematics, a frieze or frieze pattern is a design on a two-dimensional surface that is repetitive in one direction.

## Frobenius endomorphism

In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic, an important class which includes finite fields.

## Galois group

In mathematics, more specifically in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension.

## Generating set of a group

In abstract algebra, a generating set of a group is a subset such that every element of the group can be expressed as the combination (under the group operation) of finitely many elements of the subset and their inverses.

## Graph automorphism

In the mathematical field of graph theory, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving the edge–vertex connectivity.

## Greatest common divisor

In mathematics, the greatest common divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers.

## Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

## Group isomorphism

In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations.

## Hyperbolic group

In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group equipped with a word metric satisfying certain properties abstracted from classical hyperbolic geometry.

## Ideal (ring theory)

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.

## If and only if

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

## Index of a subgroup

In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" (cosets) of H that fill up G. For example, if H has index 2 in G, then intuitively half of the elements of G lie in H. The index of H in G is usually denoted |G: H| or or (G:H).

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

## Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.

## Klein four-group

In mathematics, the Klein four-group (or just Klein group or Vierergruppe, four-group, often symbolized by the letter V or as K4) is the group, the direct product of two copies of the cyclic group of order 2.

## Lattice of subgroups

In mathematics, the lattice of subgroups of a group G is the lattice whose elements are the subgroups of G, with the partial order relation being set inclusion.

## Localization of a ring

In commutative algebra, localization is a systematic method of adding multiplicative inverses to a ring.

## Locally cyclic group

In group theory, a locally cyclic group is a group (G, *) in which every finitely generated subgroup is cyclic.

## Loop (graph theory)

In graph theory, a loop (also called a self-loop or a "buckle") is an edge that connects a vertex to itself.

## Lowest common denominator

In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the lowest common multiple of the denominators of a set of fractions.

## Metacyclic group

In group theory, a metacyclic group is an extension of a cyclic group by a cyclic group.

## Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus (plural moduli).

## Modular representation theory

Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite groups over a field K of positive characteristic.

## Multigraph

In mathematics, and more specifically in graph theory, a multigraph (in contrast to a simple graph) is a graph which is permitted to have multiple edges (also called parallel edges), that is, edges that have the same end nodes.

## Multiplicative group of integers modulo n

In modular arithmetic, the integers coprime (relatively prime) to n from the set \ of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n. Equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n. Hence another name is the group of primitive residue classes modulo n. In the theory of rings, a branch of abstract algebra, it is described as the group of units of the ring of integers modulo n. Here units refers to elements with a multiplicative inverse, which in this ring are exactly those coprime to n. This group, usually denoted (\mathbb/n\mathbb)^\times, is fundamental in number theory.

## NGC 1300

NGC 1300 is a barred spiral galaxy about 61 million light-years away in the constellation Eridanus.

## Nicolas Bourbaki

Nicolas Bourbaki is the collective pseudonym under which a group of (mainly French) 20th-century mathematicians, with the aim of reformulating mathematics on an extremely abstract and formal but self-contained basis, wrote a series of books beginning in 1935.

## Nilpotent group

A nilpotent group G is a group that has an upper central series that terminates with G. Provably equivalent definitions include a group that has a central series of finite length or a lower central series that terminates with.

## Normal subgroup

In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part.

## Number theory

Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.

## Orbifold notation

In geometry, orbifold notation (or orbifold signature) is a system, invented by William Thurston and popularized by the mathematician John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature.

## Order (group theory)

In group theory, a branch of mathematics, the term order is used in two unrelated senses.

In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems.

## Path graph

In the mathematical field of graph theory, a path graph or linear graph is a graph whose vertices can be listed in the order v1, v2, …, vn such that the edges are where i.

## Polycyclic group

In mathematics, a polycyclic group is a solvable group that satisfies the maximal condition on subgroups (that is, every subgroup is finitely generated).

## Polygon

In elementary geometry, a polygon is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit.

## Prentice Hall

Prentice Hall is a major educational publisher owned by Pearson plc.

## Presentation of a group

In mathematics, one method of defining a group is by a presentation.

## Primary cyclic group

In mathematics, a primary cyclic group is a group that is both a cyclic group and a ''p''-primary group for some prime number p. That is, it has the form for some prime number p, and natural number m. Every finite abelian group G may be written as a finite direct sum of primary cyclic groups: This expression is essentially unique: there is a bijection between the sets of groups in two such expressions, which maps each group to one that is isomorphic.

## Prime ideal

In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers.

## Prime number

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

## Prime power

In mathematics, a prime power is a positive integer power of a single prime number.

## Primitive root modulo n

In modular arithmetic, a branch of number theory, a number g is a primitive root modulo n if every number a coprime to n is congruent to a power of g modulo n. That is, for every integer a coprime to n, there is an integer k such that gk ≡ a (mod n).

## Quotient group

A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure.

## Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

## Representation theory of finite groups

The representation theory of groups is a part of mathematics which examines how groups act on given structures.

## Ring (mathematics)

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.

## Ring homomorphism

In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the structure.

## Root of unity

In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power.

## Rotational symmetry

Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn.

## Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

## Simple group

In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself.

## Subgroup

In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.

## Sylow theorems

In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Ludwig Sylow (1872) that give detailed information about the number of subgroups of fixed order that a given finite group contains.

## Tensor product

In mathematics, the tensor product of two vector spaces and (over the same field) is itself a vector space, together with an operation of bilinear composition, denoted by, from ordered pairs in the Cartesian product into, in a way that generalizes the outer product.

## Triangle

A triangle is a polygon with three edges and three vertices.

## Unit (ring theory)

In mathematics, an invertible element or a unit in a (unital) ring is any element that has an inverse element in the multiplicative monoid of, i.e. an element such that The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group for this operation.

## Unit fraction

A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer.

## Up to

In mathematics, the phrase up to appears in discussions about the elements of a set (say S), and the conditions under which subsets of those elements may be considered equivalent.

## Vertex-transitive graph

In the mathematical field of graph theory, a vertex-transitive graph is a graph G such that, given any two vertices v1 and v2 of G, there is some automorphism such that In other words, a graph is vertex-transitive if its automorphism group acts transitively upon its vertices.

## Zero of a function

In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f is a member x of the domain of f such that f(x) vanishes at x; that is, x is a solution of the equation f(x).

## References

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