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+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Group Theory +-- {: .hide} [[!include group theory - contents]] =-- =-- =-- # Cyclic groups * table of contents {: toc} ## 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](#Ring)* below. ## Examples {#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|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 of a group|order]]/[[cardinality]] is [[countable set|countably]] [[infinite set|infinite]]. For $n \gt 0$, the [[order of a group|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 {#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|$p$-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](#Ring) (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 {#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 $0$). \lineabreak ### Fundamental theorem of cyclic groups {#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 [Aluffi 09, pages 83-84](#Aluffi09)) This is a special case of the _[[fundamental theorem of finitely generated abelian groups]]_. See there for more. ### Relation to finite abelian groups +-- {: .num_prop} ###### 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](http://ncatlab.org/nlab/show/projective+resolution#CohomologyOfCyclicGroups)_ * at *[[finite rotation group]]* the section *[Group cohomology](finite+rotation+group#GroupCohomology)* * in [Golasiński & Gonçalves 2011](#GolasińskiGonçalves11) 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. +-- {: .num_example #IrreducibleRealRepresentationsOfCyclicGroups} ###### Example **([[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: 1. the 1-[[dimension|dimensional]] [[trivial representation]] $\mathbf{1}$; 1. the 1-[[dimension|dimensional]] [[sign representation]] $\mathbf{1}_{sgn}$; 1. 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 $$ \rho_{\mathbf{2}_k}(1) \;=\; \left( \array{ cos(\theta) & -sin(\theta) \\ sin(\theta) & \phantom{-}cos(\theta) } \right) \phantom{AA} \text{with} \; \theta = 2 \pi \tfrac{k}{n} \,, $$ (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 [[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: $$ Rep^{irr}_{\mathbb{R}} \big( \mathbb{Z}/n \big)_{/\sim} \;=\; \big\{ \mathbf{1}, \mathbf{1}_{sgn}, \mathbf{2}_k \;\vert\; 0 \lt k \lt n/2 \big\} $$ =-- (e.g. [tom Dieck 09 (1.1.6), (1.1.8)](representation+theory#tomDieck09)) ## 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]]: * {#Klein1884} [[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](https://archive.org/details/cu31924059413439) Textbook accounts: * {#Aluffi09} [[Paolo Aluffi]], Part 0 of: *Algebra: Chapter 0*, Graduate Studies in Mathematics **104**, AMS (2009) [[ISBN:978-1-4704-1168-8](https://bookstore.ams.org/gsm-104/)] * [[Joseph A. Gallian]], Section 4 of: *Contemporary Abstract Algebra*, Chapman and Hall/CRC (2020) [[doi:10.1201/9781003142331](https://doi.org/10.1201/9781003142331), [pdf](https://ict.iitk.ac.in/wp-content/uploads/CS203-Mathematics-for-Computer-Science-III-Gallian.pdf)] Further review: * Philippe B. Laval, _Cyclic groups_ [[pdf](http://math.kennesaw.edu/~plaval/math4361/groups_cyclic.pdf)] On the [[fundamental theorem of cyclic groups]]: * {#Gallian10} 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: * {#GolasińskiGonçalves11} [[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](https://doi.org/10.1016/j.topol.2011.06.022)] [[!redirects cyclic group]] [[!redirects cyclic groups]] [[!redirects infinite cyclic group]] [[!redirects infinite cyclic groups]]