Cyclic groups

group theory

Cyclic groups

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

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

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

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

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 $\mathbb{Z}/{|n|}$.

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 ‘$\infty$’ in place of ‘$n$’ (depending on whether we think of it as $\mathbb{Z}$ modulo $0$ or the cyclic group with order $\infty$).

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 $n \gt 0$, the number of ring structures on the cyclic group $\mathbb{Z}/n$, which is the same as the number of generators, is $\phi(n)$, the Euler totient? of $n$; the generators are those $i$ that are relatively prime to $n$. While $\phi(1) = 1$, otherwise $\phi(n) \gt 1$ (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 $0$).

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Fundamental theorem of cyclic groups

For $n \in \mathbb{N}$, there is precisely one subgroup of the cyclic group $\mathbb{Z}/n\mathbb{N}$ of order $d \in \mathbb{N}$ for each factor of $d$ in $n$, and this is the subgroup generated by $n/d \in \mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$.

Moreover, the lattice of subgroups of $\mathbb{Z}/n\mathbb{Z}$ is equivalently the dual of the lattice of natural numbers $\leq n$ ordered by divisibility.

This is a special case of the fundamental theorem of finitely generated abelian groups. See there for more.

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.

Group cohomology

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

Relevant for Dijkgraaf-Witten theory is the fact

$H^3_{grp}(\mathbb{Z}/n\mathbb{Z}, U(1)) \simeq \mathbb{Z}/n\mathbb{Z} \,.$

Linear representations

We discuss some of the representation theory of cyclic grou

Example

(irreducible real linear representations of cyclic groups)

For $n \in \mathbb{N}$, $n \geq 2$, the isomorphism classes of irreducible real linear representations of the cyclic group $\mathbb{Z}/n$ are given by precisely the following:

1. the 1-dimensional trivial representation $\mathbf{1}$;

2. the 1-dimensional sign representation $\mathbf{1}_{sgn}$;

3. the 2-dimensional standard representations $\mathbf{2}_k$ of rotations in the Euclidean plane by angles that are integer multiples of $2 \pi k/n$, for $k \in \mathbb{N}$, $0 \lt k \lt n/2$;

hence the restricted representations of the defining real rep of SO(2) under the subgroup inclusions $\mathbb{Z}/n \hookrightarrow SO(2)$, hence the representations generated by real $2 \times 2$ trigonometric matrices of the form

$\rho_{\mathbf{2}_k}(1) \;=\; \left( \array{ cos(\theta) & -sin(\theta) \\ sin(\theta) & \phantom{-}cos(\theta) } \right) \phantom{AA} \text{with} \; \theta = 2 \pi \tfrac{k}{n} \,,$

(For $k = n/2$ the corresponding 2d representation is the direct sum of two copies of the sign representation: $\mathbf{2}_{n/2} \simeq \mathbf{1}_{sgn} \oplus \mathbf{1}_{sgn}$, and hence not irreducible. Moreover, for $k \gt n/2$ we have that $\mathbf{2}_{k}$ is irreducible, but isomorphic to $\mathbf{2}_{n-k} \simeq \mathbf{2}_{-k}$).

In summary:

$Rep^{irr}_{\mathbb{R}} \big( \mathbb{Z}/n \big)_{/\sim} \;=\; \big\{ \mathbf{1}, \mathbf{1}_{sgn}, \mathbf{2}_k \;\vert\; 0 \lt k \lt n/2 \big\}$

References

• Paolo Aluffi, Algebra: Chapter 0, Part 0, 2009

• Cyclic groups pdf

• Joseph A. Gallian, Fundamental Theorem of Cyclic Groups, Contemporary Abstract Algebra, p. 77, (2010)

Discussion of the cyclic group in the context of the classification of finite rotation groups:

• Felix Klein, chapter I.3 of Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade, 1884, translated as Lectures on the Icosahedron and the Resolution of Equations of Degree Five by George Morrice 1888, online version

Last revised on May 9, 2019 at 08:25:58. See the history of this page for a list of all contributions to it.