Cyclic group

In group theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

, a cyclic group is a group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

 that can be generated

Generating set of a group

In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their...

 by a single element, in the sense that the group has an element g (called a "generator

Generating set of a group

In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their...

" 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).

Definition

A group G is called cyclic if there exists an element g in G such that G = <g> = { gⁿ | n is an integer }. Since any group generated by an element in a group is a subgroup of that group, showing that the only subgroup

Subgroup

In group theory, 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 *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...

 of a group G that contains g is G itself suffices to show that G is cyclic.

For example, if G = { g⁰, g¹, g², g³, g⁴, g⁵ } is a group, then g⁶ = g⁰, and G is cyclic. In fact, G is essentially the same as (that is, isomorphic

Isomorphism

In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

 to) the set { 0, 1, 2, 3, 4, 5 } with addition modulo

Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....

 6. For example, 1 + 2 = 3 (mod 6) corresponds to g¹·g² = g³, and 2 + 5 = 1 (mod 6) corresponds to g²·g⁵ = g⁷ = g¹, and so on. One can use the isomorphism φ defined by φ(gⁱ) = i.

For every positive integer n there is exactly one cyclic group (up to isomorphism) whose order

Order (group theory)

In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....

 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 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, and is isomorphic to the additive group of integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s Z.

Furthermore, the circle group (whose elements are uncountable) is not a cyclic group—a cyclic group always has countable elements.

Since the cyclic groups are 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 their order . Abelian groups generalize the arithmetic of addition of integers...

, they are often written additively and denoted Z_n. However, this notation can be problematic for number theorists

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

 because it conflicts with the usual notation for p-adic number

P-adic number

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

 rings or localization

Localization of a ring

In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units in R*...

 at a prime ideal

Prime ideal

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

. The quotient

Quotient group

In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...

 notations Z/nZ, Z/n, and Z/(n) are standard alternatives. We adopt the first of these here to avoid the collision of notation. See also the section Subgroups and notation below.

One may write the group multiplicatively, and denote it by Cn, where n is the order (which can be ∞). For example, g³g⁴ = g² in C₅, whereas 3 + 4 = 2 in Z/5Z.

Properties

The fundamental theorem of cyclic groups

Fundamental theorem of cyclic groups

In abstract algebra, the fundamental theorem of cyclic groups states that every subgroup of a cyclic group is cyclic. Moreover, the order of any subgroup of a cyclic group G\, of order n\, is a divisor of n\,, and for each positive divisor k\, of n\, the group G\, has exactly one subgroup of order...

 states that if G is a cyclic group of order n then every subgroup

Subgroup

In group theory, 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 *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...

 of G is cyclic. Moreover, the order of any subgroup of G is a divisor of n and for each positive divisor k of n the group G has exactly one subgroup of order k. This property characterizes finite cyclic groups: a group of order n is cyclic if and only if for every divisor d of n the group has at most one subgroup of order d. Sometimes the refined statement is used: a group of order n is cyclic if and only if for every divisor d of n the group has exactly one subgroup of order d.

Every finite cyclic group is isomorphic

Isomorphism

In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

 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,

• G 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 their order . Abelian groups generalize the arithmetic of addition of integers...
  
  ; 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 = g⁰ is the identity element of the group, since kn mod n = 0 for any integer k.
• If n = ∞, then there are exactly two elements that generate the group on their own: namely 1 and −1 for Z
• If n is finite, then there are exactly φ(n) elements that generate the group on their own, where φ is the Euler totient 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/nZ (factor group of Z over nZ) since Z/nZ = {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

Divisor

In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...

 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

Greatest common divisor

In mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...

(n,m).

If p is a prime number

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

, then the only group (up to

Up to

In mathematics, the phrase "up to x" means "disregarding a possible difference in  x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...

 isomorphism

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. If there exists an isomorphism between two groups, then the groups are called isomorphic...

) with p elements is the cyclic group Cp or Z/pZ. There are more numbers with the same property, see cyclic number

Cyclic number (group theory)

A cyclic number is a natural number n such that n and φ are coprime. Here φ is Euler's totient function. An equivalent definition is that a number n is cyclic iff any group of order n is cyclic.Any prime number is clearly cyclic...

.

The direct product

Direct product of groups

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

 of two cyclic groups Z/nZ and Z/mZ is cyclic if and only if n and m are coprime

Coprime

In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...

. Thus e.g. Z/12Z is the direct product of Z/3Z and Z/4Z, but not the direct product of Z/6Z and Z/2Z.

The definition immediately implies that cyclic groups have very simple group presentation

Presentation of a group

In mathematics, one method of defining a group is by a presentation. One specifies a set S of generators so that every element of the group can be written as a product of powers of some of these generators, and a set R of relations among those generators...

 C_∞ = < x | > and C_n = < x | xⁿ > for finite n.

A primary cyclic group

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 formfor some prime number p, and natural number m....

 is a group of the form Z/p^kZ where p is a prime number

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

. The fundamental theorem of abelian groups states that every finitely generated abelian group

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 formwith integers n1,...,ns...

 is the direct product of finitely many finite primary cyclic and infinite cyclic groups.

Z/nZ and Z are also commutative ring

Commutative ring

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

s. If p is a prime, then Z/pZ is a finite field

Finite field

In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...

, also denoted by F_p or GF(p). Every field with p elements is isomorphic

Isomorphism

In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

 to this one.

The units

Unit (ring theory)

In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...

 of the ring Z/nZ are the numbers coprime

Coprime

In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...

 to n.
They form a group under multiplication modulo n

Multiplicative group of integers modulo n

In modular arithmetic the set of congruence classes relatively prime to the modulus n form a group under multiplication called the multiplicative group of integers modulo n. It is also called the group of primitive residue classes modulo n. In the theory of rings, a branch of abstract algebra, it...

 with φ(n) elements (see above). It is written as (Z/nZ)^×.
For example, when n = 6, we get (Z/nZ)^× = {1,5}.
When n = 8, we get (Z/nZ)^× = {1,3,5,7}.

In fact, it is known that (Z/nZ)^× is cyclic if and only if n is 1 or 2 or 4 or pk or 2 pk for an odd prime number

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

 p and k ≥ 1, in which case every generator of (Z/nZ)^× is called a primitive root modulo n

Primitive root modulo n

In modular arithmetic, a branch of number theory, a primitive root modulo n is any number g with the property that any number coprime to n is congruent to a power of g modulo n. In other words, g is a generator of the multiplicative group of integers modulo n...

. Thus, (Z/nZ)^× is cyclic for n = 6, but not for n = 8, where it is instead isomorphic to the Klein four-group

Klein four-group

In mathematics, the Klein four-group is the group Z2 × Z2, the direct product of two copies of the cyclic group of order 2...

.

The group (Z/pZ)^× is cyclic with p − 1 elements for every prime p, and is also written (Z/pZ)^* because it consists of the non-zero elements. More generally, every finite subgroup

Subgroup

In group theory, 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 *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...

 of the multiplicative group of any field

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

 is cyclic.

Examples

In 2D and 3D the symmetry group

Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...

 for n-fold rotational symmetry

Rotational symmetry

Generally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation. An object may have more than one rotational symmetry; for instance, if reflections or turning it over are not counted, the triskelion appearing on the Isle of Man's flag has...

 is Cn, of abstract group type Z_n. In 3D there are also other symmetry groups which are algebraically the same, see Symmetry groups in 3D that are cyclic as abstract group.

Note that the group S¹ of all rotations of a circle

Circle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

 (the circle group) is not cyclic, since it is not even countable.

The n^th roots of unity

Root of unity

In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...

 form a cyclic group of order n under multiplication. e.g., where and a group of under multiplication is cyclic.

The Galois group

Galois group

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

 of every finite field extension

Field extension

In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...

 of a finite field

Finite field

In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...

 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.

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.

C1	C2	C3	C4	C5	C6	C7	C8

The representation theory

Representation theory of finite groups

In mathematics, representation theory is a technique for analyzing abstract groups in terms of groups of linear transformations. See the article on group representations for an introduction...

 of the cyclic group is a critical base case for the representation theory of more general finite groups. In the complex case

Character theory

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

, a representation of a cyclic group decomposes into a direct sum of linear characters, making the connection between character theory and representation theory transparent. In the positive characteristic case

Modular representation theory

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

, the indecomposable representations of the cyclic group form a model and inductive basis for the representation theory of groups with cyclic Sylow subgroups and more generally the representation theory of blocks of cyclic defect.

Subgroups and notation

All subgroup

Subgroup

In group theory, 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 *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...

s and quotient group

Quotient group

In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...

s 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

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....

 of Z is isomorphic to the dual

Duality (order theory)

In the mathematical area of order theory, every partially ordered set P gives rise to a dual partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the set with the inverse order, i.e. x ≤ y holds in Pop if and only if y ≤ x holds in P...

 of the lattice of natural numbers ordered by divisibility. All factor groups of Z are finite, except for the trivial exception Z/{0} = Z/0Z. 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

Simple group

In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...

 if and only if its order (the number of its elements) is prime.

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

Ring (mathematics)

In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

 terminology, the subgroup nZ is also the ideal

Ideal (ring theory)

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....

 (n), so the quotient can also be written Z/(n) or Z/n without abuse of notation. These alternatives do not conflict with the notation for the p-adic integers. The last form is very common in informal calculations; it has the additional advantage that it reads the same way that the group or ring is often described verbally in English, "Zee mod en".

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 gk 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 greatest common divisor

Greatest common divisor

In mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...

 of k and n. Put another way, the index

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" 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...

 of the subgroup generated by gk is m. This reasoning is known as the index calculus algorithm, in number theory

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

.

Endomorphisms

The endomorphism ring

Endomorphism ring

In abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object; this may be denoted End...

 of the abelian group Z/nZ is isomorphic

Ring homomorphism

In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication....

 to Z/nZ itself as a ring. Under this isomorphism, the number r corresponds to the endomorphism of Z/nZ that 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/nZ is isomorphic to the unit group (Z/nZ)^× (see above).

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.

Virtually cyclic groups

A group is called virtually cyclic if it contains a cyclic subgroup of finite index (the number of coset

Coset

In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G...

s that the subgroup has).
In other words, any element in a virtually cyclic group can be arrived at by applying a member of the cyclic subgroup to a member in a certain finite set.
Every cyclic group is virtually cyclic, as is every finite group.
It is known that a finitely generated discrete group

Discrete group

In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...

 with exactly two ends is virtually cyclic (for instance the product

Direct product of groups

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

 of Z/n and Z). Every abelian subgroup of a Gromov hyperbolic group

Hyperbolic group

In group theory, a hyperbolic group, also known as a word hyperbolic group, Gromov hyperbolic group, negatively curved group is a finitely generated group equipped with a word metric satisfying certain properties characteristic of hyperbolic geometry. The notion of a hyperbolic group was introduced...

 is virtually cyclic.

See also

• Cyclic extension
• Cyclic module
  Cyclic module
  In mathematics, more specifically in ring theory, a cyclic module is a module over a ring which is generated by one element. The term is by analogy with cyclic groups, that is groups which are generated by one element.- Definition :...
  
  
• Cyclically ordered group
  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....
  
  
• Locally cyclic group
  Locally cyclic group
  In group theory, a locally cyclic group is a group in which every finitely generated subgroup is cyclic.-Some facts:*Every cyclic group is locally cyclic, and every locally cyclic group is abelian....
  
  , a group in which each finitely generated subgroup is cyclic
• Modular arithmetic
  Modular arithmetic
  In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
  
  

External links

• An introduction to cyclic groups