[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).