nLab cyclic group

Definition

A cyclic group is a quotient group of the additive group of integers (hence of the free group on the singleton).

Generally one considers cyclic groups as abstract groups, that is without specifying which element comprises the generating singleton. But see at Ring structure below.

Examples

There is (up to isomorphism) one cyclic group

$$\mathbb{Z}\!/\!n \,\coloneqq\, \mathbb{Z}/n\mathbb{Z}$$

for every natural number $n \in \mathbb{N}$, this being the quotient group by the (necessarily normal) subgroup $n \mathbb{Z} \hookrightarrow \mathbb{Z}$ of integers divisible by $n$.

For $n = 0$ this subgroup is the trivial group, hence this quotient is just the integers itself

$$\mathbb{Z}\!/\!0 \,\simeq\, \mathbb{Z} \,,$$

as such also called the infinite cyclic group, since its order/cardinality is countably infinite.

For $n \gt 0$, the order (cardinality) of $\mathbb{Z}\!/\!n$ is the finite number

$$ord\big( \mathbb{Z}\!/\!n \big) \;=\; card\big( \mathbb{Z}\!/\!n \big) \;=\; n \,.$$

Explicitly this means that

• elements of $\mathbb{Z}\!/\!n$ are equivalence classes $[k]$ of integers, where the equivalence relation is “modulo $n$”:

  $$[k] = [k'] \;\;\; \in \; \mathbb{Z}\!/\!n \;\;\;\;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\;\;\;\; \underset{r \in \mathbb{Z}}{\exists} \big( k ' = k + r \cdot n \big) \,.$$

• the binary operation in the group is addition of representatives

  $$\array{ \mathbb{Z}\!/\!n \times \mathbb{Z}\!/\!n &\longrightarrow& \mathbb{Z}\!/\!n \\ \big( [k_1], \, [k_2] \big) &\mapsto& [k_1 + k_2] \,. }$$

The cyclic group $\mathbb{Z}\!/\!n$ of order $n$ may also be identified with a subgroup of the multiplicative group of units $\mathbb{C}^\times \hookrightarrow \mathbb{C}$ of complex numbers (or algebraic numbers), namely the group of $n$th roots of unity:

$$\array{ \mathbb{Z}\!/\!n &\xhookrightarrow{\phantom{---}}& \mathbb{C}^\times \\ [k] &\mapsto& \exp\big(2 \pi \mathrm{i} \tfrac{k}{n}\big) \mathrlap{\,.} }$$

Dedicated entries exist for the examples of the

• cyclic group of order 2

• cyclic group of order 3

• cyclic group of order 4

Notation

Some alternative notations for the finite cyclic groups are in use. Many authors use subscript notation

• “$\mathbb{Z}_n$” for $\mathbb{Z}\!/\! n \mathbb{Z}$.

However, at least for $n = p$ a prime number, this notation clashes with standard notation “$\mathbb{Z}_p$” for the ring of $p$-adic integers.

Therefore, in fields where both cyclic groups as well as p-adic integers play an important role (such as in algebraic topology and arithmetic geometry, see e.g. the theory of cyclotomic spectra), it is common to choose different notation for the cyclic groups, typically

• “$C_n$” for $\mathbb{Z}\!/\! n \mathbb{Z}$.

Here “$C$” is, of course, for “cyclic”. Other authors may keep the letter “Z” with a subscript but resort to another font, such as

• “$Z_n$” for $\mathbb{Z}\!/\! n \mathbb{Z}$.

Often this last notation is meant to indicate that not just the group but the ring-structure inherited from $\mathbb{Z}$ is referred to, see below (which of course makes the possible confusion with notation for the p-adic integers only worse).

Incidentally, while the notation “$\mathbb{Z}$” for the integers derives from the German word Zahl (for number), that letter happens to also be the first one in the German word zyklisch (for cyclic).

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 $0$).

\lineabreak

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.

(e.g Aluffi 09, pages 83-84)

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 discussion of the group cohomology of cyclic groups see

• at projective resolution the section Cohomology of cyclic groups

• at finite rotation group the section Group cohomology

• in Golasiński & Gonçalves 2011 for the cases $H^n_{Grp}(/mathbb{Z}/n;\, \mathbb{Z}/m)$

For example, relevant for Dijkgraaf-Witten theory is the fact:

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

Linear representations

We discuss some of the representation theory of cyclic groups.

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:

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

(e.g. tom Dieck 09 (1.1.6), (1.1.8))

Related concepts

• group of order 2

• binary cyclic group

• permutation cycle, cyclic permutation

• multiplicative group of integers modulo n

• Q/Z

• integers modulo n

• p-localization

• profinite completion of the integers

• cyclotomic field

• cyclotomic spectrum

References

Historical 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

Textbook accounts:

• Paolo Aluffi, Part 0 of: Algebra: Chapter 0, Graduate Studies in Mathematics 104, AMS (2009) [ISBN:978-1-4704-1168-8]

• Joseph A. Gallian, Section 4 of: Contemporary Abstract Algebra, Chapman and Hall/CRC (2020) [doi:10.1201/9781003142331, pdf]

Further review:

• Philippe B. Laval, Cyclic groups [pdf]

On the fundamental theorem of cyclic groups:

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

On the group cohomology of cyclic groups with coefficients in cyclic groups:

• Marek Golasiński, Daciberg Lima Gonçalves, On cohomologies and extensions of cyclic groups, Topology and its Applications 158 (2011) 1858–1865 [doi:10.1016/j.topol.2011.06.022]