In mathematics, a

**cyclic group**is a group that can be generated by a single element, in the sense that the group has an element

*a*(called a "generator" of the group) such that all elements of the group are powers of

*a*. Equivalently, an element

*a*of a group

*G*generates

*G*precisely if

*G*is the only subgroup of itself that contains

*a*.

The cyclic groups are the simplest groups and they are completely known: for any positive integer *n*, there is a cyclic group *C*_{n} of order *n*, and then there is the infinite cyclic group, the additive group of integers **Z**.
Every other cyclic group is isomorphic to one of these.

The finite cyclic groups can be introduced as a series of symmetry groups, or as the groups of rotations of a regular n-gon: for example C_{3} can be represented as the group of rotations of an equilateral triangle. While this example is concise and graphical, it is important to remember that each element of C_{3} represent an *action* and not a position. Note also that the group *S*^{1} of all rotations of a circle is *not* cyclic.

The cyclic group *C*_{n} is isomorphic to the group **Z**/*n***Z** of integers modulo *n* with addition as operation; an isomorphism is given by the discrete logarithm. One typically writes the group *C*_{n} multiplicatively, while **Z**/*n***Z** is written additively. Sometimes **Z**_{n} is used instead of **Z**/*n***Z**.

## Properties

All cyclic groups are abelian, that is they are commutative.

The element *a* mentioned above in the definition is called a *generator* of the cyclic group. A cyclic group can have several generators. The generators of **Z** are +1 and -1, the generators of **Z**/*n***Z** are the residue classes of the integers which are coprime to *n*; the number of those generators is known as φ(*n*), where φ is Euler's phi function.

More generally, if *d* is a divisor of *n*, then the number of elements in **Z**/*n***Z** which have order *d* is φ(*d*). The order of the residue class of *m* is *n* / gcd(*n*,*m*).

If *p* is a prime number, then the only group (up to isomorphism) with *p* elements is the cyclic group *C*_{p}.

The direct product of two cyclic groups *C*_{n} and *C*_{m} is cyclic if and only if *n* and *m* are coprime.

Every finitely generated abelian group is the direct product of finitely many cyclic groups.

All subgroups and factor groups of cyclic groups are cyclic. Specifically, the subgroups of **Z** are of the form *m***Z**, with *m* a natural number. All these subgroups are different, and the non-zero ones are all 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 group **Z**/*n***Z** 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 the number of its elements is prime.

The endomorphism ring of the abelian group *C*_{n} is isomorphic to the ring **Z**/*n***Z**. Under this isomorphism, the residue class of *r* in **Z**/*n***Z** corresponds to the endomorphism of *C*_{n} which raises every element to the *r*-th power. As a consequence, the automorphism group of *C*_{n} is isomorphic to the group (**Z**/*n***Z**)^{×}, the group of units of the ring **Z**/*n***Z**. This is the group of numbers coprime to *n* under multiplication modulo *n*; it has φ(*n*) elements.

Similarly, the endomorphism ring of the infinite cyclic group is isomorphic to the ring **Z**, and its automorphism group is isomorphic to the group of units of the ring **Z**, i.e. to {-1, +1} ≅ *C*_{2}.

## Advanced examples

If *n* is a positive integer, then (**Z**/*n***Z**)^{×} 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. The generators of this cyclic group are called primitive roots modulo *n*.

In particular, the group (**Z**/*p***Z**)^{×} is cyclic with *p* -1 elements for every prime *p*. More generally, every *finite* subgroup of the multiplicative group of any field is cyclic.

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*.