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\begin{document}

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\section*{cyclic group}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

\hypertarget{cyclic_groups}{}\section*{{Cyclic groups}}\label{cyclic_groups}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{notation}{Notation}\dotfill \pageref*{notation} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{Ring}{Ring structure}\dotfill \pageref*{Ring} \linebreak
\noindent\hyperlink{FundamentalTheoremOfCyclicGroups}{Fundamental theorem of cyclic groups}\dotfill \pageref*{FundamentalTheoremOfCyclicGroups} \linebreak
\noindent\hyperlink{relation_to_finite_abelian_groups}{Relation to finite abelian groups}\dotfill \pageref*{relation_to_finite_abelian_groups} \linebreak
\noindent\hyperlink{group_cohomology}{Group cohomology}\dotfill \pageref*{group_cohomology} \linebreak
\noindent\hyperlink{linear_representations}{Linear representations}\dotfill \pageref*{linear_representations} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A \textbf{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 \hyperlink{Ring}{Ring structure} below.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

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

\begin{displaymath}
\mathbb{Z}_n \coloneqq \mathbb{Z}/n\mathbb{Z}
  \,.
\end{displaymath}
For $n \gt 0$, the [[order of a group|order]] ([[cardinality]]) of $\mathbb{Z}_n$ is $n$ (so [[finite set|finite]]); for $n = 0$, which is the group of [[integers]]

\begin{displaymath}
\mathbb{Z}_0 \coloneqq \mathbb{Z}
\end{displaymath}
the [[order of a group|order]] is [[countable set|countable]] but [[infinite set|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 \emph{not} a root of $1$ (but there is no standard choice) and take the subgroup generated by it.

Dedicated entries exist for:

\begin{itemize}%
\item [[cyclic group of order 2]]


\item [[cyclic group of order 3]]


\item [[cyclic group of order 4]]



\end{itemize}
\hypertarget{notation}{}\subsection*{{Notation}}\label{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 \emph{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.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{Ring}{}\subsubsection*{{Ring structure}}\label{Ring}

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 [[equivalence of categories|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 [[totient function|Euler totient]] of $n$; the generators are those $i$ that are [[relatively prime numbers|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 $0$).

$\backslash$lineabreak

\hypertarget{FundamentalTheoremOfCyclicGroups}{}\subsubsection*{{Fundamental theorem of cyclic groups}}\label{FundamentalTheoremOfCyclicGroups}

For $n \in \mathbb{N}$, there is precisely one [[subgroup]] of the cyclic group $\mathbb{Z}/n\mathbb{N}$ of [[order of a group|order]] $d \in \mathbb{N}$ for each factor of $d$ in $n$, and this is the subgroup [[generators and relations|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 \hyperlink{Aluffi09}{Aluffi 09, pages 83-84})

This is a special case of the \emph{[[fundamental theorem of finitely generated abelian groups]]}. See there for more.

\hypertarget{relation_to_finite_abelian_groups}{}\subsubsection*{{Relation to finite abelian groups}}\label{relation_to_finite_abelian_groups}

\begin{prop}
\label{}\hypertarget{}{}
Every [[finite abelian group]] is a [[direct sum of abelian groups]] over cyclic groups.

\end{prop}
See at \emph{[[finite abelian group]]} for details.

\hypertarget{group_cohomology}{}\subsubsection*{{Group cohomology}}\label{group_cohomology}

For a discussion of the [[group cohomology]] of cyclic groups see at \emph{[[projective resolution]]} in the section \emph{\href{http://ncatlab.org/nlab/show/projective+resolution#CohomologyOfCyclicGroups}{Cohomology of cyclic groups}}.

Relevant for [[Dijkgraaf-Witten theory]] is the fact

\begin{displaymath}
H^3_{grp}(\mathbb{Z}/n\mathbb{Z}, U(1)) \simeq \mathbb{Z}/n\mathbb{Z}
  \,.
\end{displaymath}
\hypertarget{linear_representations}{}\subsubsection*{{Linear representations}}\label{linear_representations}

We discuss some of the [[representation theory]] of cyclic grou

\begin{example}
\label{IrreducibleRealRepresentationsOfCyclicGroups}\hypertarget{IrreducibleRealRepresentationsOfCyclicGroups}{}
\textbf{([[irreducible representation|irreducible]] [[real numbers|real]] [[linear representations]] of [[cyclic groups]])}

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

\begin{enumerate}%
\item the 1-[[dimension|dimensional]] [[trivial representation]] $\mathbf{1}$;


\item the 1-[[dimension|dimensional]] [[sign representation]] $\mathbf{1}_{sgn}$;


\item the 2-[[dimension|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 number|real]] $2 \times 2$ [[trigonometric function|trigonometric]] [[matrices]] of the form

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


\end{enumerate}
(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 [[irrep|irreducible]]. Moreover, for $k \gt n/2$ we have that $\mathbf{2}_{k}$ is irreducible, but [[isomorphism|isomorphic]] to $\mathbf{2}_{n-k} \simeq \mathbf{2}_{-k}$).

In summary:

\begin{displaymath}
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\}
\end{displaymath}
\end{example}
(e.g. \href{representation+theory#tomDieck09}{tom Dieck 09 (1.1.6), (1.1.8)})

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[group of order 2]]


\item [[binary cyclic group]]


\item [[multiplicative group of integers modulo n]]


\item [[cyclic ring]]


\item [[p-localization]]


\item [[profinite completion of the integers]]


\item [[cyclotomic field]]


\item [[cyclotomic spectrum]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Paolo Aluffi, \emph{Algebra: Chapter 0, Part 0}, 2009


\item \emph{Cyclic groups} \href{http://math.kennesaw.edu/~plaval/math4361/groups_cyclic.pdf}{pdf}


\item Joseph A. Gallian, \emph{Fundamental Theorem of Cyclic Groups}, Contemporary Abstract Algebra, p. 77, (2010)



\end{itemize}
Discussion of the cyclic group in the context of the [[classification of finite rotation groups]]:

\begin{itemize}%
\item [[Felix Klein]], chapter I.3 of \emph{Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade}, 1884, translated as \emph{Lectures on the Icosahedron and the Resolution of Equations of Degree Five} by George Morrice 1888, \href{https://archive.org/details/cu31924059413439}{online version}

\end{itemize}
[[!redirects cyclic group]] [[!redirects cyclic groups]]

[[!redirects cyclic group of order 2]]



\end{document}