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 $n\in Z$ such that $h=g^n$, where we have written $g^n$ for $g+g+\cdots +g$ (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 $G=⟨g⟩$ if $g$ is a generator of $G$.

Examples

• $(Z,+)=⟨1⟩=⟨-1⟩$

• 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).