nLab cyclic group

Cyclic groups

Cyclic groups


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.


There is (up to isomorphism) one cyclic group

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

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

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

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

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

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

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

Explicitly this means that

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

    [k]=[k]/nr(k=k+rn). [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

    /n×/n /n ([k 1],[k 2]) [k 1+k 2]. \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 /n\mathbb{Z}\!/\!n of order nn 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 nnth roots of unity:

/n × [k] exp(2πikn). \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


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

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

However, at least for n=pn = p a prime number, this notation clashes with standard notation “ p\mathbb{Z}_p” for the ring of p 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 nC_n” for /n\mathbb{Z}\!/\! n \mathbb{Z}.

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

  • Z nZ_n” for /n\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).


Ring structure

Let AA be a cyclic group, and let xx be a generator of AA. Then there is a unique ring structure on AA (making the original group the additive group of the ring) such that xx is the multiplicative identity 11.

If we identify AA with the additive group /n\mathbb{Z}/n and pick (the equivalence class of) the integer 11 for xx, then the resulting ring is precisely the quotient ring /n\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>0n \gt 0, the number of ring structures on the cyclic group /n\mathbb{Z}/n, which is the same as the number of generators, is ϕ(n)\phi(n), the Euler totient? of nn; the generators are those ii that are relatively prime to nn. While ϕ(1)=1\phi(1) = 1, otherwise ϕ(n)>1\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 11 and 1-1 are the generators (and these are relatively prime to 00).


Fundamental theorem of cyclic groups

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

Moreover, the lattice of subgroups of /n\mathbb{Z}/n\mathbb{Z} is equivalently the dual of the lattice of natural numbers n\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


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

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

H grp 3(/n,U(1))/n. 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.


(irreducible real linear representations of cyclic groups)

For nn \in \mathbb{N}, n2n \geq 2, the isomorphism classes of irreducible real linear representations of the cyclic group /n\mathbb{Z}/n are given by precisely the following:

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

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

  3. the 2-dimensional standard representations 2 k\mathbf{2}_k of rotations in the Euclidean plane by angles that are integer multiples of 2πk/n2 \pi k/n, for kk \in \mathbb{N}, 0<k<n/20 \lt k \lt n/2;

    hence the restricted representations of the defining real rep of SO(2) under the subgroup inclusions /nSO(2)\mathbb{Z}/n \hookrightarrow SO(2), hence the representations generated by real 2×22 \times 2 trigonometric matrices of the form

    ρ 2 k(1)=(cos(θ) sin(θ) sin(θ) cos(θ))AAwithθ=2πkn, \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/2k = n/2 the corresponding 2d representation is the direct sum of two copies of the sign representation: 2 n/21 sgn1 sgn\mathbf{2}_{n/2} \simeq \mathbf{1}_{sgn} \oplus \mathbf{1}_{sgn}, and hence not irreducible. Moreover, for k>n/2k \gt n/2 we have that 2 k\mathbf{2}_{k} is irreducible, but isomorphic to 2 nk2 k\mathbf{2}_{n-k} \simeq \mathbf{2}_{-k}).

In summary:

Rep irr(/n) /={1,1 sgn,2 k|0<k<n/2} 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))


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:

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:

Last revised on April 26, 2024 at 10:00:07. See the history of this page for a list of all contributions to it.