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).