nLab
cyclic group

Definition

A cyclic group is a quotient group of the free group on the singleton.

Generally, we consider a cyclic group as a group, that is without specifying which element comprises the generating singleton. But see Ring structure below.

Examples

There is (up to isomorphism) one cyclic group for every natural number $n$, denoted

$${\mathbb{Z}}_n≔\mathbb{Z}/n\mathbb{Z}.$$\mathbb{Z}_n \coloneqq \mathbb{Z}/n\mathbb{Z} \,.

For $n>0$, the order (cardinality) of ${\mathbb{Z}}_n$ is $n$ (so finite); for $n=0$, which is the group of integers

$${\mathbb{Z}}_0≔\mathbb{Z}$$\mathbb{Z}_0 \coloneqq \mathbb{Z}

the order is countable but infinite.

If we identify the free group on the singleton with the additive group $\mathbb{Z}$ of integers, then the infinite cyclic group is $\mathbb{Z}$ itself, while the finite cyclic group of order $n$ is $\mathbb{Z}/n$, that is $\mathbb{Z}$ modulo (the normal subgroup generated by) the integer $n$. Of course, $\mathbb{Z}$ itself is also $\mathbb{Z}/0$. One could also consider $\mathbb{Z}/n$ for negative $n$, but this is the same as $\mathbb{Z}/\mid n\mid$.

The cyclic group of order $n$ may also be identified with a subgroup of the multiplicative group of complex numbers (or algebraic numbers): the group of $n$th roots of $1$. For $n=0$, we may pick any non-zero complex number (or even something else) that is not a root of $1$ (but there is no standard choice) and take the subgroup generated by it.

Notation

Besides ‘$\mathbb{Z}/n$’, the cyclic group of order $n$ may also be denoted in other ways: some more complicated variation of ‘$\mathbb{Z}/n$’ (to put ‘the [normal] subgroup generated by’ explicitly in the notation), or else the simplified form ‘${\mathbb{Z}}_n$’ (which however conflicts with notation for the $n$-adic integers). When written multiplicatively, it may be denoted ‘$Z_n$’ (note the font change) or ‘$C_n$’; either letter here stands for ‘cyclic’ in one language or another. (It is a coincidence that the German words ‘Zahl’, which gives us ‘$\mathbb{Z}$’, and ‘zyklisch’, which gives us ‘$Z$’, begin with the same letter.)

Besides ‘$\mathbb{Z}$’, the infinite cyclic group may also be denoted in other ways: some variation of ‘$(\mathbb{Z},+)$’ to indicate that we are using addition of integers, or any of the above notations with either ‘$0$’ or ‘$\mathrm{\infty }$’ in place of ‘$n$’ (depending on whether we think of it as $\mathbb{Z}$ modulo $0$ or the cyclic group with order $\mathrm{\infty }$).

When written additively, the notation for the elements of a cyclic group are usually just the notation for integers; for the finite cyclic group of order $n$, we use the natural number less than $n$. In the finite case, we may also use brackets or some other notation to indicate equivalence classes. When written multiplicatively, any letter (‘$e$’, ‘$x$’, ‘$a$’, ‘$\xi$’, etc) may be taken to stand for the generating element; then any other element is a power of this generator. When thought of as a multiplicative group of complex numbers, one generator is ${\mathrm{e}}^{2\mathrm{i}\pi /n}$, and the notation may reflect that.

Properties

Ring structure

Let $A$ be a cyclic group, and let $x$ be a generator of $A$. Then there is a unique ring structure on $A$ (making the original group the additive group of the ring) such that $x$ is the multiplicative identity $1$.

If we identify $A$ with the additive group $\mathbb{Z}/n$ and pick (the equivalence class of) the integer $1$ for $x$, then the resulting ring is precisely the quotient ring? $\mathbb{Z}/n$.

In this way, a cyclic group equipped with the extra structure of a generator is the same thing (in the sense that their groupoids are equivalent) as a ring with the extra property that the underlying additive group is cyclic.

For $n>0$, the number of ring structures on the cyclic group $\mathbb{Z}/n$, which is the same as the number of generators, is $\varphi (n)$, the Euler totient? of $n$; the generators are those $i$ that are relatively prime to $n$. While $\varphi (1)=1$, otherwise $\varphi (n)>1$ (another way to see that we have a structure and not just a property). For $\mathbb{Z}$ itself, there are two ring structures, since $1$ and $-1$ are the generators (and these are relatively prime to $0$).

Group cohomology

For a discussion of the group cohomology of cyclic groups see at projective resolution in the section Cohomology of cyclic groups.

Relation to finite abelian groups

Proposition

Every finite abelian group is a direct sum of abelian groups over cyclic groups.

See at finite abelian group for details.

Related concepts

• group of order 2