A group is said to be cyclic if it is generated by a single element.

Suppose $G$ is a cyclic group generated by $x\in G$ . Then every element of $G$ is equal to $x^k$ for some $k\in \Z$ . If $G$ is infinite, then these $x^k$ are all distinct, and $G$ is isomorphic to the group $\Z$ . If $G$ has finite order $n$ , then every element of $G$ can be expressed as $x^k$ with $k\in\{0,\dots,n-1\}$ , and $G$ is isomorphic to the quotient group $\Z/n\Z$ .

Note that the isomorphisms mentioned in the previous paragraph imply that all cyclic groups of the same order are isomorphic to one another. The infinite cyclic group is sometimes written $C_\infty$ , and the finite cyclic group of order $n$ is sometimes written $C_n$ . However, when the cyclic groups are written additively, they are commonly represented by $\Z$ and $\Z/n\Z$ .

While a cyclic group can, by definition, be generated by a single element, there are often a number of different elements that can be used as the generator: an infinite cyclic group has $2$ generators, and a finite cyclic group of order $n$ has $\phi(n)$ generators, where $\phi$ is the Euler totient function.

Some basic facts about cyclic groups:

• Every cyclic group is abelian.
• Every subgroup of a cyclic group is cyclic.
• Every quotient of a cyclic group is cyclic.
• Every group of prime order is cyclic. (This follows immediately from Lagrange's Theorem.)
• Every finite subgroup of the multiplicative group of a field is cyclic.