In group theory, a cyclic group or monogenous group is a group that can be generated by a single element, in the sense that the group has an element g (called a "generator" of the group) such that, when written multiplicatively, every element of the group is a power of g (a multiple of g when the notation is additive). Group theory is that branch of mathematics concerned with the study of groups. ... This picture illustrates how the hours on a clock form a group under modular addition. ... In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ... In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ...

Definition

A group G is called cyclic if there exists an element g in G such that G = <g> = { gⁿ for all integers n }. Since any group generated by an element in a group is a subgroup of that group, showing that the only subgroup of a group G that contains g is G itself suffices to show that G is cyclic. In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...

For example, if G = { e, g¹, g², g³, g⁴, g⁵ }, then G is cyclic, and, G is essentially the same as (that is, isomorphic to) the set { 0, 1, 2, 3, 4, 5 } with addition modulo 6. I.e. 1 + 2 mod 6 = 3, 2 + 5 mod 6 = 1, and so on. One can use the isomorphism φ defined by φ(g) = 1. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ...

For every positive integer n there is exactly one cyclic group (up to isomorphism) whose order is n, and there is exactly one infinite cyclic group (the integers under addition). Hence, the cyclic groups are the simplest groups and they are completely classified.

The name 'cyclic' may be misleading: it is possible to generate infinitely many elements and not form any literal cycles; that is, every gⁿ is distinct (It can be said that it has one infinitely long cycle). A group generated in this way is called an infinite cyclic group, which is isomorphic to the additive group of integers . The integers are commonly denoted by the above symbol. ...

Since the cyclic groups are abelian, they are often written additively and denoted Z_n. However, this notation can be problematic for number theorists because it conflicts with the usual notation for p-adic number rings or localization at a prime ideal. The quotient notation Z/n or Z/nZ (see also below) is a standard alternative, which we adopt here to avoid the collision of notation. In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... In mathematics, the p-adic number systems were first described by Kurt Hensel in 1897. ... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that collapses the normal subgroup N to the identity element. ...

One may write the group multiplicatively, and denote it by C_n. (For example, g³g⁴ = g² in C₅, whereas 3 + 4 = 2 in Z/5.)

An alternative definition for a finite cyclic group is as follows.

A group of order is cyclic if for every divisor of the group has at most one subgroup of order .

Properties

The fundamental theorem of cyclic groups states that if is a cyclic group of order then every subgroup of is cyclic. Moreover, the order of any subgroup of is a divisor of and for each positive divisor of the group has exactly one subgroup of order . In abstract algebra, the fundamental theorem of cyclic groups states that if is a cyclic group of order then every subgroup of is cyclic. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...

Every finite cyclic group is isomorphic to the group { [0], [1], [2], ..., [n − 1] } of integers modulo n under addition, and any infinite cyclic group is isomorphic to Z (the set of all integers) under addition. Thus, one only needs to look at such groups to understand the properties of cyclic groups in general. Hence, cyclic groups are one of the simplest groups to study and a number of nice properties are known. Given a cyclic group G of order n (n may be infinity) and for every g in G, In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...

• G is abelian; that is, their group operation is commutative: gh = hg (for all h in G). This is so since g + h mod n = h + g mod n.
• If n is finite, then gⁿ = e since n mod n = 0.
• If n = ∞, then there are exactly two generators: namely 1 and −1 for Z, and any others mapped to them under an isomorphism in other infinite cyclic groups.
• If n is finite, then there are exactly φ(n) generators where φ() is the Euler phi function
• Every subgroup of G is cyclic. Indeed, each finite subgroup of G is a group of { 0, 1, 2, 3, ... m − 1} with addition modulo m. And each infinite subgroup of G is mZ for some m, which is bijective to (so isomorphic to) Z.
• G_n is isomorphic to Z/n (factor group of Z over nZ) since Z/n = {0 + nZ, 1 + nZ, 2 + nZ, 3 + nZ, 4 + nZ, ..., n − 1 + nZ} { 0, 1, 2, 3, 4, ..., n − 1} under addition modulo n.

More generally, if d is a divisor of n, then the number of elements in Z/n which have order d is φ(d). The order of the residue class of m is n / gcd(n,m). In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ... In number theory, the totient &#966;(n) of a positive integer n is defined to be the number of positive integers less than or equal to n and coprime to n. ... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively collapses the normal subgroup N to the identity element. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ...

If p is a prime number, then the only group (up to isomorphism) with p elements is the cyclic group C_p or Z/p. In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ... 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. ...

The direct product of two cyclic groups Z/n and Z/m is cyclic if and only if n and m are coprime. Thus e.g. Z/12 is the direct product of Z/3 and Z/4, but not the direct product of Z/6 and Z/2. In mathematics, one can often define a direct product of objects already known, giving a new one. ... In mathematics, the integers a and b are said to be coprime or relatively prime if they have no common factor other than 1 and −1, or equivalently, if their greatest common divisor is 1. ...

The definition immediately implies that cyclic groups have very simple group presentation C_n = < x | xⁿ >. In mathematics, one method of defining a group is by a presentation. ...

The fundamental theorem of abelian groups states that every finitely generated abelian group is the direct product of finitely many finite cyclic and infinite cyclic groups. In abstract algebra, an abelian group (G,+) 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 x = n1x1 + n2x2 + ... + nsxs with integers n1,...,ns. ... In abstract algebra, an abelian group (G,+) 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 x = n1x1 + n2x2 + ... + nsxs with integers n1,...,ns. ...

Z/n and Z are also commutative rings. If p is a prime, then Z/p is a finite field, also denoted by F_p or GF(p). Every field with p elements is isomorphic to this one. In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...

The units of the ring Z/n are the numbers coprime to n. They form a group under multiplication modulo n with φ(n) elements (see above). It is written as (Z/n)^×. For example, we get (Z/n)^× = {1,5} when n = 6, and get (Z/n)^× = {1,3,5,7} when n = 8. In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of... In mathematics, the integers a and b are said to be coprime or relatively prime if they have no common factor other than 1 and −1, or equivalently, if their greatest common divisor is 1. ... In mathematics, the multiplicative group of integers modulo n is the group defined by multiplication of the units (that is, the numbers relatively prime to ) in the ring for a given integer . ...

In fact, it is known that (Z/n)^× is cyclic if and only if n is 2 or 4 or p^k or 2 p^k for an odd prime number p and k ≥ 1, in which case every generator of (Z/n)^× is called a primitive root modulo n. Thus, (Z/n)^× is cyclic for n = 6, but not for n = 8, where it is instead isomorphic to the Klein four-group. In mathematics, any integer (whole number) is either even or odd. ... In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... A primitive root modulo n is a concept from modular arithmetic in number theory. ... This article is about the mathematical group. ...

The group (Z/p)^× is cyclic with p − 1 elements for every prime p, and is also written (Z/p)^* because it consists of the non-zero elements. More generally, every finite subgroup of the multiplicative group of any field is cyclic. In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

Examples

In 2D and 3D the symmetry group for n-fold rotational symmetry is C_n, of abstract group type Z_n. In 3D there are also other symmetry groups which are algebraically the same, see Cyclic symmetry groups in 3D. The symmetry group of an object (e. ... The triskelion appearing on the Isle of Man flag. ... A discrete point group in 3D is a finite symmetry group in 3D that leaves the origin fixed. ...

Note that the group S¹ of all rotations of a circle (the circle group) is not cyclic, since it is not even countable. Circle illustration This article is about the shape and mathematical concept of circle. ... In mathematics, the circle group, denoted by T (or in blackboard bold by ), is the multiplicative group of all complex numbers with absolute value 1. ... In mathematics the term countable set is used to describe the size of a set, e. ...

The n^th roots of unity form a cyclic group of order n under multiplication. e.g., 0 = z³ − 1 = (z − s⁰)(z − s¹)(z − s²) where sⁱ = e^{2πi / 3} and a group of {s⁰,s¹,s²} under multiplication is cyclic. In mathematics, the nth roots of unity, or de Moivre numbers, are all the complex numbers which yield 1 when raised to a given power n. ...

The Galois group of every finite field extension of a finite field is finite and cyclic; conversely, given a finite field F and a finite cyclic group G, there is a finite field extension of F whose Galois group is G. In mathematics, a Galois group is a group associated with a certain type of field extension. ... In mathematics, more specifically in abstract algebra, field extensions are the main object of study in field theory. ... In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. ...

Representation

The cycle graphs of finite cyclic groups are all n-sided polygons with the elements at the vertices. The dark vertex in the cycle graphs below stand for the identity element, and the other vertices are the other elements of the group. A cycle consists of successive powers of either of the elements connected to the identity element. In group theory a group cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups. ...

C1	C2	C3	C4	C5	C6	C7	C8

Cycle diagam for the group C1 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... Image File history File links GroupDiagramMiniC2. ... Cycle diagram for group C3 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... Cycle diagram for group C4 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... Cycle diagram for group C5 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... Cycle diagram for group C6 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... Cycle diagram for group C7 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... Cycle diagram for group C8 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ...

Subgroups and notation

All subgroups and quotient groups of cyclic groups are cyclic. Specifically, all subgroups of Z are of the form mZ, with m an integer ≥0. All of these subgroups are different, and apart from the trivial group (for m=0) all are isomorphic to Z. The lattice of subgroups of Z is isomorphic to the dual of the lattice of natural numbers ordered by divisibility. All factor groups of Z are finite, except for the trivial exception Z / {0}. For every positive divisor d of n, the quotient group Z/nZ has precisely one subgroup of order d, the one generated by the residue class of n/d. There are no other subgroups. The lattice of subgroups is thus isomorphic to the set of divisors of n, ordered by divisibility. In particular, a cyclic group is simple if and only if its order (the number of its elements) is prime. In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that collapses the normal subgroup N to the identity element. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... The lattice of subgroups of the dihedral group Dih4, represented as groups of rotations and reflections of a plane figure. ... 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. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... In mathematics, a simple group is a group which is not the trivial group and whose only normal subgroups are the trivial group and the group itself. ...

Using the quotient group formalism, Z/nZ is a standard notation for the additive cyclic group with n elements.

In ring terminology, the subgroup nZ is also the ideal (n), so the quotient can also be written Z/(n) or Z/n without abuse of notation. The last form has the advantages that it reads exactly the same way that the group or ring is often described verbally, "Zee mod en"; and it does not conflict with the notation for p-adic integers. In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...

As a practical problem, one may be given a finite subgroup C of order n, generated by an element g, and asked to find the size m of the subgroup generated by g^k for some integer k. Here m will be the smallest integer > 0 such that mk is divisible by n. It is therefore n/m where m = (k, n) is the gcd of k and n. Put another way, the index of the subgroup generated by g^k is m. This reasoning is known as the index calculus algorithm, in number theory. The three letter acronym GCD may refer to: Greatest common divisor — in mathematics Great circle distance — in navigation Griffith College Dublin — private college in Ireland Grand Comic-Book Database — database of comic book information Global Communications Devices — supplier of semiconductor devices used in wireless networking Gardner Carton & Douglas — a US... In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G... In group theory, the index calculus algorithm is an algorithm for computing discrete logarithms. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ...

Endomorphisms

The endomorphism ring of the abelian group Z/n is isomorphic to Z/n itself as a ring. Under this isomorphism, the number r corresponds to the endomorphism of Z/n which maps each element to the sum of r copies of it. This is a bijection if and only if r is coprime with n, so the automorphism group of Z/n is isomorphic to the unit group (Z/n)^× (see above). In abstract algebra, one associates to certain objects a ring, the objects endomorphism ring, which encodes several internal properties of the object. ... In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ... In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ...

Similarly, the endomorphism ring of the additive group Z is isomorphic to the ring Z. Its automorphism group is isomorphic to the group of units of the ring Z, i.e. to {−1, +1} C₂.

See also

• Cyclic extension
• Cyclic module
• Modular arithmetic