[summary: The cyclic groups are the simplest kind of group; they are the groups which can be made by simply "repeating a single element many times". For example, the rotations of a polygon.]

[summary(technical): A group $~$G$~$ is cyclic if it has a single [generator_mathematics generator]: there is one element $~$g$~$ such that every element of the group is a [ power] of $~$g$~$.]

Definition

A cyclic group is a group $~$(G, +)$~$ (hereafter abbreviated as simply $~$G$~$) with a single generator, in the sense that there is some $~$g \in G$~$ such that for every $~$h \in G$~$, there is $~$n \in \mathbb{Z}$~$ such that $~$h = g^n$~$, where we have written $~$g^n$~$ for $~$g + g + \dots + g$~$ (with $~$n$~$ terms in the summand). That is, "there is some element such that the group has nothing in it except powers of that element".

We may write $~$G = \langle g \rangle$~$ if $~$g$~$ is a generator of $~$G$~$.

Examples

• $~$(\mathbb{Z}, +) = \langle 1 \rangle = \langle -1 \rangle$~$
• The group with two elements (say $~$\{ e, g \}$~$ with identity $~$e$~$ with the only possible group operation $~$g^2 = e$~$) is cyclic: it is generated by the non-identity element. Note that there is no requirement that the powers of $~$g$~$ be distinct: in this case, $~$g^2 = g^0 = e$~$.
• The integers modulo $~$n$~$ form a cyclic group under addition, for any $~$n$~$: it is generated by $~$1$~$ (or, indeed, by $~$n-1$~$).
• The symmetric groups $~$S_n$~$ for $~$n > 2$~$ are not cyclic. This can be deduced from the fact that they are not abelian (see below).

Properties

Cyclic groups are abelian

Suppose $~$a, b \in G$~$, and let $~$g$~$ be a generator of $~$G$~$. Suppose $~$a = g^i, b = g^j$~$. Then $~$ab = g^i g^j = g^{i+j} = g^{j+i} = g^j g^i = ba$~$.

Cyclic groups are countable

The elements of a cyclic group are nothing more nor less than $~$\{ g^0, g^1, g^{-1}, g^2, g^{-2}, \dots \}$~$, which is an enumeration of the group (possibly with repeats).