Editing Cyclic group

{{short description|Mathematical group that can be generated as the set of powers of a single element}}
{{Group theory sidebar |Finite}}
In [[abstract algebra]], a '''cyclic group''' or '''monogenous group''' is a [[Group (mathematics)|group]], denoted C<sub>''n''</sub> (also frequently '''<math>\Z</math>'''<sub>''n''</sub> or Z<sub>''n''</sub>, not to be confused with the commutative ring of [[P-adic number|{{mvar|p}}-adic numbers]]), that is [[Generating set of a group|generated]] by a single element.<ref name="eom">{{springer|title=Cyclic group|id=p/c027510}}</ref> That is, it is a [[set (mathematics)|set]] of [[Inverse element|invertible]] elements with a single [[associative]] [[binary operation]], and it contains an element&nbsp;''g'' such that every other element of the group may be obtained by repeatedly applying the group operation to&nbsp;''g'' or its inverse. Each element can be written as an integer [[Exponentiation|power]] of&nbsp;''g'' in multiplicative notation, or as an integer multiple of ''g'' in additive notation. This element&nbsp;''g'' is called a ''[[Generating set of a group|generator]]'' of the group.<ref name="eom"/>

Every infinite cyclic group is [[isomorphic]] to the [[additive group]] of&nbsp;'''Z''', the [[integer]]s. Every finite cyclic group of [[Order (group theory)|order]]&nbsp;''n'' is isomorphic to the additive group of [[Quotient group|'''Z'''/''n'''''Z''']], the integers [[modular arithmetic|modulo]]&nbsp;''n''. Every cyclic group is an [[abelian group]] (meaning that its group operation is [[commutative property|commutative]]), and every [[finitely generated group|finitely generated]] abelian group is a [[Direct product of groups|direct product]] of cyclic groups.

Every cyclic group of [[prime number|prime]] order is a [[simple group]], which cannot be broken down into smaller groups. In the [[classification of finite simple groups]], one of the three infinite classes consists of the cyclic groups of prime order. The cyclic groups of prime order are thus among the building blocks from which all groups can be built.

==Definition and notation==
[[File:Cyclic group.svg|left|thumb|160px|The six 6th complex [[Root of unity|roots of unity]] form a cyclic group under multiplication. Here, ''z'' is a generator, but ''z''<sup>2</sup> is not, because its powers fail to produce the odd powers of&nbsp;''z''.]]

For any element&nbsp;''g'' in any group&nbsp;''G'', one can form the [[subgroup]] that consists of all its integer [[Exponentiation|powers]]: {{nowrap|1=⟨''g''⟩ = {{mset| ''g''<sup>''k''</sup> {{!}} ''k'' ∈ '''Z''' }}}}, called the '''cyclic subgroup''' generated by&nbsp;''g''. The [[Order (group theory)|order]] of&nbsp;''g'' is |⟨''g''⟩|, the number of elements in&nbsp;⟨''g''⟩, conventionally abbreviated as |''g''|, as ord(''g''), or as o(''g''). That is, the order of an element is equal to the order of the cyclic subgroup that it generates.

A ''cyclic group'' is a group which is equal to one of its cyclic subgroups: {{nowrap|1=''G'' = ⟨''g''⟩}} for some element&nbsp;''g'', called a [[Generating set of a group|''generator'']] of ''G''.

For a '''finite cyclic group'''&nbsp;''G'' of order&nbsp;''n'' we have {{nowrap|1=''G'' = {{mset|''e'', ''g'', ''g''<sup>2</sup>, ... , ''g''<sup>''n''−1</sup>}}}}, where ''e'' is the identity element and {{nowrap|1=''g''<sup>''i''</sup> = ''g''<sup>''j''</sup>}} whenever {{nowrap|''i'' ≡ ''j''}} ([[modular arithmetic|mod]]&nbsp;''n'');  in particular {{nowrap|1=''g''<sup>''n''</sup> = ''g''<sup>0</sup> = ''e''}}, and {{nowrap|1=''g''<sup>−1</sup> = ''g''<sup>''n''&minus;1</sup>}}. An abstract group defined by this multiplication is often denoted C<sub>''n''</sub>, and we say that ''G'' is [[Isomorphism|isomorphic]] to the standard cyclic group&nbsp;C<sub>''n''</sub>. Such a group is also isomorphic to '''Z'''/''n'''''Z''', the group of integers modulo ''n'' with the addition operation, which is the standard cyclic group in additive notation. Under the isomorphism ''&chi;'' defined by {{nowrap|1=''&chi;''(''g''<sup>''i''</sup>) = ''i''}} the identity element&nbsp;''e'' corresponds to&nbsp;0, products correspond to sums, and powers correspond to multiples.

For example, the set of complex 6th roots of unity: <math display="block">G = \left\{\pm 1, \pm{ \left(\tfrac 1 2 + \tfrac{\sqrt 3}{2}i\right)}, \pm{\left(\tfrac 1 2 - \tfrac{\sqrt 3}{2}i\right)}\right\}</math> forms a group under multiplication. It is cyclic, since it is generated by the [[Root of unity#General definition|primitive root]] <math>z = \tfrac 1 2 + \tfrac{\sqrt 3}{2}i=e^{2\pi i/6}:</math> that is, ''G'' = ⟨''z''⟩ = { 1, ''z'', ''z''<sup>2</sup>, ''z''<sup>3</sup>, ''z''<sup>4</sup>, ''z''<sup>5</sup> } with ''z''<sup>6</sup> = 1. Under a change of letters, this is isomorphic to (structurally the same as) the standard cyclic group of order 6, defined as C<sub>6</sub> = ⟨''g''⟩ = {{mset| ''e'', ''g'', ''g''<sup>2</sup>, ''g''<sup>3</sup>, ''g''<sup>4</sup>, ''g''<sup>5</sup> }} with multiplication ''g''<sup>''j''</sup> · ''g''<sup>''k''</sup> = ''g''<sup>''j''+''k''</sup> <sup>(mod 6)</sup>, so that ''g''<sup>6</sup> = ''g''<sup>0</sup> = ''e''.  These groups are also isomorphic to '''Z'''/6'''Z''' = {{mset|0, 1, 2, 3, 4, 5}} with the operation of addition [[modular arithmetic|modulo]] 6, with ''z''<sup>''k''</sup> and ''g''<sup>''k''</sup> corresponding to&nbsp;''k''. For example, {{nowrap|1 + 2 ≡ 3 (mod 6)}} corresponds to {{nowrap|1=''z''<sup>1</sup> · ''z''<sup>2</sup> = ''z''<sup>3</sup>}}, and {{nowrap|2 + 5 ≡ 1 (mod 6)}} corresponds to {{nowrap|1=''z''<sup>2</sup> · ''z''<sup>5</sup> = ''z''<sup>7</sup> = ''z''<sup>1</sup>}}, and so on. Any element generates its own cyclic subgroup, such as ⟨''z''<sup>2</sup>⟩ = {{mset| ''e'', ''z''<sup>2</sup>, ''z''<sup>4</sup> }} of order 3, isomorphic to C<sub>3</sub> and '''Z'''/3'''Z'''; and ⟨''z''<sup>5</sup>⟩ = { ''e'', ''z''<sup>5</sup>, ''z''<sup>10</sup> = ''z''<sup>4</sup>, ''z''<sup>15</sup> = ''z''<sup>3</sup>, ''z''<sup>20</sup> = ''z''<sup>2</sup>, ''z''<sup>25</sup> = ''z'' } = ''G'', so that ''z''<sup>5</sup> has order 6 and is an alternative generator of&nbsp;''G''.

Instead of the [[quotient group|quotient]] notations '''Z'''/''n'''''Z''', '''Z'''/(''n''), or '''Z'''/''n'', some authors denote a finite cyclic group as '''Z'''<sub>''n''</sub>, but this clashes with the notation of [[number theory]], where '''Z'''<sub>''p''</sub> denotes a [[p-adic number|''p''-adic number]] ring, or [[localization of a ring|localization]] at a [[prime ideal]].

{| class=wikitable align=left width=240 style="margin-right: 20px;"
|+ Infinite cyclic groups
!p1, ([[Orbifold notation|*∞∞]])
!p11g, (22∞)
|-
|[[File:Frieze group 11.png|120px]]
|[[File:Frieze group 1g.png|120px]]
|- valign=top
|[[File:Frieze example p1.png|120px]]<br>[[File:Frieze hop.png|120px]]
|[[File:Frieze example p11g.png|120px]]<br>[[File:Frieze step.png|120px]]
|-
|colspan=2|Two [[frieze group]]s are isomorphic to&nbsp;'''Z'''. With one generator, p1 has translations and p11g has glide reflections.
|}
On the other hand, in an '''infinite cyclic group''' {{nowrap|1=''G'' = ⟨''g''⟩}}, the powers ''g''<sup>''k''</sup> give distinct elements for all integers ''k'', so that ''G'' = {{mset| ... , ''g''<sup>&minus;2</sup>, ''g''<sup>&minus;1</sup>, ''e'', ''g'', ''g''<sup>2</sup>, ... }}, and ''G'' is isomorphic to the standard group {{nowrap|1=C = C<sub>∞</sub>}} and to '''Z''', the additive group of the integers. An example is the first [[Frieze group#Descriptions of the seven frieze groups|frieze group]]. Here there are no finite cycles, and the name "cyclic" may be misleading.<ref>{{Harv|Lajoie|Mura|2000|pp=29–33}}.</ref>

To avoid this confusion, [[Nicolas Bourbaki|Bourbaki]] introduced the term '''monogenous group''' for a group with a single generator and restricted "cyclic group" to mean a finite monogenous group, avoiding the term "infinite cyclic group".{{refn|group=note|name="algebra1.§4.10"|DEFINITION 15. ''A group is called'' monogenous ''if it admits a system of generators consisting of a single element. A finite monogenous group is called'' cyclic.<ref>{{Harv|Bourbaki|1998|p=49}} or {{Google books|STS9aZ6F204C|Algebra I: Chapters 1–3|page=49}}.</ref>}}
{{Clear}}

==Examples==
{| class=wikitable align=right
|+Examples of cyclic groups in rotational symmetry
|-
| [[File:Triangle.Scalene.svg|120px]]
| [[File:Hubble2005-01-barred-spiral-galaxy-NGC1300.jpg|120px]]
| [[File:The armoured triskelion on the flag of the Isle of Man.svg|120px]]
|-
![[Scalene triangle|C<sub>1</sub>]]
![[NGC 1300|C<sub>2</sub>]]
![[Flag of the Isle of Man|C<sub>3</sub>]]
|-
|[[File:Circular-cross-decorative-knot-12crossings.svg|120px]]
| [[File:Flag of Hong Kong.svg|120px]]
|[[File:Olavsrose.svg|120px]]
|-
![[Celtic knot|C<sub>4</sub>]]
![[Flag of Hong Kong|C<sub>5</sub>]]
![[:de:Olavsrose|C<sub>6</sub>]]
|}

===Integer and modular addition===
The set of [[integer]]s&nbsp;'''Z''', with the operation of addition, forms a group.<ref name="eom"/> It is an '''infinite cyclic group''', because all integers can be written by repeatedly adding or subtracting the single number&nbsp;1. In this group, 1 and −1 are the only generators. Every infinite cyclic group is isomorphic to&nbsp;'''Z'''.

For every positive integer&nbsp;''n'', the set of integers [[modular arithmetic|modulo]]&nbsp;''n'', again with the operation of addition, forms a finite cyclic group, denoted '''Z'''/''n'''''Z'''.<ref name="eom"/>
A modular integer&nbsp;''i'' is a generator of this group if ''i'' is [[relatively prime]] to&nbsp;''n'', because these elements can generate all other elements of the group through integer addition.
(The number of such generators is ''φ''(''n''), where ''φ'' is the [[Euler totient function]].)
Every finite cyclic group ''G'' is isomorphic to '''Z'''/''n'''''Z''', where ''n'' = {{abs|''G''}} is the order of the group.

The addition operations on integers and modular integers, used to define the cyclic groups, are the addition operations of [[commutative ring]]s, also denoted '''Z''' and '''Z'''/''n'''''Z''' or '''Z'''/(''n''). If ''p'' is a [[prime number|prime]], then '''Z'''/''p'''Z''''' is a [[finite field]], and is usually denoted '''F'''<sub>''p''</sub> or GF(''p'') for Galois field.

===Modular multiplication===
{{main|Multiplicative group of integers modulo n}}
For every positive integer&nbsp;''n'', the set of the integers modulo&nbsp;''n'' that are relatively prime to&nbsp;''n'' is written as ('''Z'''/''n'''''Z''')<sup>×</sup>; it [[Multiplicative group of integers modulo n|forms a group]] under the operation of multiplication. This group is not always cyclic, but is so whenever ''n'' is 1, 2, 4, a [[prime power|power of an odd prime]], or twice a power of an odd prime {{OEIS|A033948}}.<ref>{{Harv|Motwani|Raghavan|1995|p=401}}.</ref><ref>{{Harv|Vinogradov|2003|loc=§ VI PRIMITIVE ROOTS AND INDICES|pp=105–132}}.</ref>
This is the multiplicative group of [[Unit (ring theory)|units]] of the ring '''Z'''/''n'''''Z'''; there are ''φ''(''n'') of them, where again ''φ'' is the [[Euler totient function]]. For example, ('''Z'''/6'''Z''')<sup>×</sup> = {{mset|1, 5}}, and since 6 is twice an odd prime this is a cyclic group. In contrast, ('''Z'''/8'''Z''')<sup>×</sup> = {{mset|1, 3, 5, 7}} is a [[Klein group|Klein 4-group]] and is not cyclic. When ('''Z'''/''n'''''Z''')<sup>×</sup> is cyclic, its generators are called [[primitive root modulo n|primitive roots modulo ''n'']].

For a prime number&nbsp;''p'', the group ('''Z'''/''p'''''Z''')<sup>×</sup> is always cyclic, consisting of the non-zero elements of the [[finite field]] of order&nbsp;''p''. More generally, every finite [[subgroup]] of the multiplicative group of any [[field (mathematics)|field]] is cyclic.<ref>{{Harv|Rotman|1998|p=65}}.</ref>

===Rotational symmetries===
{{main|Rotational symmetry}}
The set of [[rotational symmetry|rotational symmetries]] of a [[polygon]] forms a finite cyclic group.<ref>{{Harv|Stewart|Golubitsky|2010|pp=47–48}}.</ref> If there are ''n'' different ways of moving the polygon to itself by a rotation (including the null rotation) then this symmetry group is isomorphic to '''Z'''/''n'''''Z'''. In three or higher dimensions there exist other [[Point groups in three dimensions#Cyclic 3D symmetry groups|finite symmetry groups that are cyclic]], but which are not all rotations around an axis, but instead [[rotoreflection]]s.

The group of all rotations of a [[circle]] (the [[circle group]], also denoted ''S''<sup>1</sup>) is ''not'' cyclic, because there is no single rotation whose integer powers generate all rotations. In fact, the infinite cyclic group C<sub>∞</sub> is [[countable]], while ''S''<sup>1</sup> is not. The group of rotations by rational angles ''is'' countable, but still not cyclic.

===Galois theory===
An ''n''th [[root of unity]] is a [[complex number]] whose ''n''th power is&nbsp;1, a [[root of a polynomial|root]] of the [[polynomial]] {{nowrap|''x''<sup>''n''</sup> − 1}}. The set of all ''n''th roots of unity forms a cyclic group of order ''n'' under multiplication.<ref name="eom"/> The generators of this cyclic group are the ''n''th [[primitive root of unity|primitive roots of unity]]; they are the roots of the ''n''th [[cyclotomic polynomial]].
For example, the polynomial {{nowrap|1=''z''<sup>3</sup> − 1}} factors as {{nowrap|(''z'' − 1)(''z'' − ''ω'')(''z'' − ''ω''<sup>2</sup>)}}, where {{nowrap|1=''ω'' = ''e''<sup>2''πi''/3</sup>}}; the set {{mset|1, ''ω'', ''ω''<sup>2</sup>}} = {{mset|''ω''<sup>0</sup>, ''ω''<sup>1</sup>, ''ω''<sup>2</sup>}} forms a cyclic group under multiplication. The [[Galois group]] of the [[field extension]] of the [[rational number]]s generated by the ''n''th roots of unity forms a different group, isomorphic to the multiplicative group ('''Z/'''''n'''''Z''')<sup>×</sup> of order [[Euler's totient function|''φ''(''n'')]], which is cyclic for some but not all&nbsp;''n'' (see above).

A field extension is called a [[cyclic extension]] if its Galois group is cyclic. For fields of [[Characteristic (algebra)|characteristic zero]], such extensions are the subject of [[Kummer theory]], and are intimately related to [[solvability by radicals]]. For an extension of [[finite field]]s of characteristic&nbsp;''p'', its Galois group is always finite and cyclic, generated by a power of the [[Frobenius endomorphism|Frobenius mapping]].<ref>{{Harv|Cox|2012|loc = Theorem&nbsp;11.1.7|p=294}}.</ref> Conversely, given a finite field&nbsp;''F'' and a finite cyclic group&nbsp;''G'', there is a finite field extension of&nbsp;''F'' whose Galois group is&nbsp;''G''.<ref>{{Harv|Cox|2012|loc=Corollary 11.1.8 and Theorem 11.1.9|p=295}}.</ref>

==Subgroups==
{{main|Fundamental theorem of cyclic groups}}
All [[subgroup]]s and [[quotient group]]s of cyclic groups are cyclic. Specifically, all subgroups of '''Z''' are of the form ⟨''m''⟩ = ''m'''''Z''', with ''m'' a positive integer. All of these subgroups are distinct from each other, and apart from the trivial group {0} = 0'''Z''', they all are [[isomorphic]] to&nbsp;'''Z'''. The [[lattice of subgroups]] of '''Z''' is isomorphic to the [[Duality (order theory)|dual]] of the lattice of natural numbers ordered by [[divisibility]].<ref>{{Harv|Aluffi|2009|pp=82–84|loc=6.4 Example: Subgroups of Cyclic Groups}}.</ref> Thus, since a prime number&nbsp;''p'' has no nontrivial divisors, ''p'''''Z''' is a maximal proper subgroup, and the quotient group '''Z'''/''p'''''Z''' is [[simple group|simple]]; in fact, a cyclic group is simple if and only if its order is prime.<ref>{{Harv|Gannon|2006|p=18|}}.</ref>

All quotient groups '''Z'''/''n'''''Z''' are finite, with the exception {{nowrap|1='''Z'''/0'''Z''' = '''Z'''/{0}.}} For every positive divisor&nbsp;''d'' of&nbsp;''n'', the quotient group '''Z'''/''n'''''Z''' has precisely one subgroup of order&nbsp;''d'', generated by the [[Modular arithmetic#Congruence classes|residue class]] of&nbsp;''n''/''d''. There are no other subgroups.

==Additional properties==
Every cyclic group is [[abelian group|abelian]].<ref name="eom"/> That is, its group operation is [[Commutative property|commutative]]: {{nowrap|1=''gh'' = ''hg''}} (for all ''g'' and ''h'' in ''G''). This is clear for the groups of integer and modular addition since {{nowrap|''r'' + ''s'' ≡ ''s'' + ''r'' (mod ''n'')}}, and it follows for all cyclic groups since they are all isomorphic to these standard groups. For a finite cyclic group of order ''n'', ''g''<sup>''n''</sup> is the identity element for any element&nbsp;''g''. This again follows by using the isomorphism to modular addition, since {{nowrap|''kn'' ≡ 0 (mod ''n'')}} for every integer&nbsp;''k''. (This is also true for a general group of order ''n'', due to [[Lagrange's theorem (group theory)|Lagrange's theorem]].)

For a [[prime power]] <math>p^k</math>, the group <math>Z/p^k Z</math> is called a [[primary cyclic group]]. The [[fundamental theorem of finitely generated abelian groups|fundamental theorem of abelian groups]] states that every [[finitely generated abelian group]] is a finite direct product of primary cyclic and infinite cyclic groups.

Because a cyclic group is abelian, each of its [[conjugacy class]]es consists of a single element. A cyclic group of order&nbsp;''n'' therefore has ''n'' conjugacy classes.

If ''d'' is a [[divisor]] of&nbsp;''n'', then the number of elements in '''Z'''/''n'''''Z''' which have order ''d'' is ''φ''(''d''), and the number of elements whose order divides ''d'' is exactly&nbsp;''d''.
If ''G'' is a finite group in which, for each {{nowrap|''n'' > 0}}, ''G'' contains at most ''n'' elements of order dividing ''n'', then ''G'' must be cyclic.{{refn|group=note|This implication remains true even if only prime values of ''n'' are considered.<ref>{{Harv|Gallian|2010|loc = Exercise&nbsp;43|p=84}}.</ref> (And observe that when ''n'' is prime, there is exactly one element whose order is a proper divisor of&nbsp;''n'', namely the identity.)}}
The order of an element ''m'' in '''Z'''/''n'''''Z''' is ''n''/[[greatest common divisor|gcd]](''n'',''m'').

If ''n'' and ''m'' are [[coprime]], then the [[direct product of groups|direct product]] of two cyclic groups '''Z'''/''n'''''Z''' and '''Z'''/''m'''''Z''' is isomorphic to the cyclic group '''Z'''/''nm'''''Z''', and the converse also holds: this is one form of the [[Chinese remainder theorem]]. For example, '''Z'''/12'''Z''' is isomorphic to the direct product {{nowrap|'''Z'''/3'''Z''' × '''Z'''/4'''Z'''}} under the isomorphism {{nowrap|(''k'' mod 12) → (''k'' mod 3, ''k'' mod 4)}}; but it is not isomorphic to {{nowrap|'''Z'''/6'''Z''' × '''Z'''/2'''Z'''}}, in which every element has order at most&nbsp;6.

If ''p'' is a [[prime number]], then any group with ''p'' elements is isomorphic to the simple group '''Z'''/''p'''''Z'''.
A number ''n''  is called a [[cyclic number (group theory)|cyclic number]] if '''Z'''/''n'''''Z''' is the only group of order&nbsp;''n'', which is true exactly when {{nowrap|1=gcd(''n'', ''φ''(''n'')) = 1}}.<ref>{{Harv|Jungnickel|1992|pp=545–547}}.</ref> The sequence of cyclic numbers include all primes, but some are [[composite number|composite]] such as&nbsp;15. However, all cyclic numbers are odd except&nbsp;2. The cyclic numbers are:

:1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 123, 127, 131, 133, 137, 139, 141, 143, ... {{OEIS|id=A003277}}

The definition immediately implies that cyclic groups have [[presentation of a group|group presentation]] {{nowrap|1=C<sub>∞</sub> = {{langle}}''x'' {{!}} {{rangle}}}} and {{nowrap|1=C<sub>''n''</sub> = {{langle}}''x'' {{!}} ''x''<sup>''n''</sup>{{rangle}}}} for finite&nbsp;''n''.<ref>{{Harv|Coxeter|Moser|1980|p=1}}.</ref>

==Associated objects==

===Representations===
The [[Representation theory of finite groups|representation theory]] of the cyclic group is a critical base case for the representation theory of more general finite groups.  In the [[character theory|complex case]], a representation of a cyclic group decomposes into a direct sum of linear characters, making the connection between character theory and representation theory transparent.  In the [[modular representation theory|positive characteristic case]], the indecomposable representations of the cyclic group form a model and inductive basis for the representation theory of groups with cyclic [[Sylow subgroup]]s and more generally the representation theory of blocks of cyclic defect.

===Cycle graph===
{{Further|Cycle graph (algebra)}}
{{anchor|Z_sub2}}
A '''cycle graph''' illustrates the various cycles of a [[group (mathematics)|group]] and is particularly useful in visualizing the structure of small [[finite group]]s. A cycle graph for a cyclic group is simply a [[circular graph]], where the group order is equal to the number of nodes. A single generator defines the group as a directional path on the graph, and the inverse generator defines a backwards path. A trivial path (identity) can be drawn as a [[Loop (graph theory)|loop]] but is usually suppressed. Z<sub>2</sub> is sometimes drawn with two curved edges as a [[multigraph]].<ref>{{MathWorld|title=Cycle Graph|id=CycleGraph}}</ref>

A cyclic group Z<sub>''n''</sub>, with order ''n'', corresponds to a single cycle graphed simply as an ''n''-sided polygon with the elements at the vertices.
{| class="wikitable"
|+ Cycle graphs up to order 24
|- align=center
| [[File:GroupDiagramMiniC1.svg|40px]]
| [[File:GroupDiagramMiniC2.svg|60px]]
| [[File:GroupDiagramMiniC3.svg|60px]]
| [[File:GroupDiagramMiniC4.svg|60px]]
| [[File:GroupDiagramMiniC5.svg|60px]]
| [[File:GroupDiagramMiniC6.svg|60px]]
| [[File:GroupDiagramMiniC7.svg|60px]]
| [[File:GroupDiagramMiniC8.svg|60px]]
|- align=center
| Z<sub>1</sub>|| Z<sub>2</sub>|| Z<sub>3</sub>|| Z<sub>4</sub>|| Z<sub>5</sub>|| Z<sub>6</sub> = Z<sub>3</sub>×Z<sub>2</sub>|| Z<sub>7</sub>|| Z<sub>8</sub>
|- align=center
| [[File:GroupDiagramMiniC9.svg|60px]]
| [[File:GroupDiagramMiniC10.svg|60px]]
| [[File:GroupDiagramMiniC11.svg|60px]]
| [[File:GroupDiagramMiniC12.svg|60px]]
| [[File:GroupDiagramMiniC13.svg|60px]]
| [[File:GroupDiagramMiniC14.svg|60px]]
| [[File:GroupDiagramMiniC15.svg|60px]]
| [[File:GroupDiagramMiniC16.svg|60px]]
|- align=center
| Z<sub>9</sub>||Z<sub>10</sub> = Z<sub>5</sub>×Z<sub>2</sub>||Z<sub>11</sub>||Z<sub>12</sub> = Z<sub>4</sub>×Z<sub>3</sub>||Z<sub>13</sub>||Z<sub>14</sub> = Z<sub>7</sub>×Z<sub>2</sub>||Z<sub>15</sub> = Z<sub>5</sub>×Z<sub>3</sub>||Z<sub>16</sub>
|- align=center
| [[File:GroupDiagramMiniC17.svg|60px]]
| [[File:GroupDiagramMiniC18.svg|60px]]
| [[File:GroupDiagramMiniC19.svg|60px]]
| [[File:GroupDiagramMiniC20.svg|60px]]
| [[File:GroupDiagramMiniC21.svg|60px]]
| [[File:GroupDiagramMiniC22.svg|60px]]
| [[File:GroupDiagramMiniC23.svg|60px]]
| [[File:GroupDiagramMiniC24.svg|60px]]
|- align=center
| Z<sub>17</sub>||Z<sub>18</sub> = Z<sub>9</sub>×Z<sub>2</sub>||Z<sub>19</sub>||Z<sub>20</sub> = Z<sub>5</sub>×Z<sub>4</sub>||Z<sub>21</sub> = Z<sub>7</sub>×Z<sub>3</sub>||Z<sub>22</sub> = Z<sub>11</sub>×Z<sub>2</sub>||Z<sub>23</sub>||Z<sub>24</sub> = Z<sub>8</sub>×Z<sub>3</sub>
|}

===Cayley graph===
{{main|Circulant graph}}
[[File:Paley13.svg|thumb|240px|The [[Paley graph]] of order 13, a circulant graph formed as the Cayley graph of '''Z'''/13 with generator set {1,3,4}]]
A [[Cayley graph]] is a graph defined from a pair (''G'',''S'') where ''G'' is a group and ''S'' is a set of generators for the group; it has a vertex for each group element, and an edge for each product of an element with a generator. In the case of a finite cyclic group, with its single generator, the Cayley graph is a [[cycle graph]], and for an infinite cyclic group with its generator the Cayley graph is a doubly infinite [[path graph]]. However, Cayley graphs can be defined from other sets of generators as well. The Cayley graphs of cyclic groups with arbitrary generator sets are called [[circulant graph]]s.<ref>{{Harv|Alspach|1997|pp=1–22}}.</ref> These graphs may be represented geometrically as a set of equally spaced points on a circle or on a line, with each point connected to neighbors with the same set of distances as each other point. They are exactly the [[vertex-transitive graph]]s whose [[Graph automorphism|symmetry group]] includes a transitive cyclic group.<ref>{{Harv|Vilfred|2004|pp=34–36}}.</ref>

===Endomorphisms===
The [[endomorphism ring]] of the abelian group '''Z'''/''n'''''Z''' is [[ring homomorphism|isomorphic]] to '''Z'''/''n'''''Z''' itself as a [[ring (algebra)|ring]].<ref name="ks">{{Harv|Kurzweil|Stellmacher|2004|p=50}}.</ref> Under this isomorphism, the number ''r'' corresponds to the endomorphism of '''Z'''/''n'''''Z''' that maps each element to the sum of ''r'' copies of it. This is a [[bijection]] if and only if ''r'' is coprime with ''n'', so the [[automorphism group]] of '''Z'''/''n'''''Z''' is isomorphic to the unit group ('''Z'''/''n'''''Z''')<sup>×</sup>.<ref name="ks"/>

Similarly, the endomorphism ring of the additive group of&nbsp;'''Z''' is isomorphic to the ring&nbsp;'''Z'''.  Its automorphism group is isomorphic to the group of units of the ring&nbsp;'''Z''', which is {{nowrap|({−1, +1}, ×) ≅ C<sub>2</sub>}}.

===Tensor product and Hom of cyclic groups===
The [[tensor product]] {{nowrap|'''Z'''/''m'''''Z''' ⊗ '''Z'''/''n'''''Z'''}} can be shown to be isomorphic to {{nowrap|'''Z''' / gcd(''m'', ''n'')'''Z'''}}. So we can form the collection of group [[Group homomorphism|homomorphisms]] from '''Z'''/''m'''''Z''' to '''Z'''/''n'''''Z''', denoted {{nowrap|hom('''Z'''/''m'''''Z''', '''Z'''/''n'''''Z''')}}, which is itself a group.

For the tensor product, this is a consequence of the general fact that {{nowrap|''R''/''I'' ⊗<sub>''R''</sub> ''R''/''J'' ≅ ''R''/(''I'' + ''J'')}}, where ''R'' is a commutative [[Ring (mathematics)|ring]] with unit and ''I'' and ''J'' are [[Ideal (ring theory)|ideals]] of the ring. For the Hom group, recall that it is isomorphic to the subgroup of {{nowrap|'''Z''' / ''n'''''Z'''}} consisting of the elements of order dividing ''m''. That subgroup is cyclic of order {{nowrap|gcd(''m'', ''n'')}}, which completes the proof.

==Related classes of groups==
Several other classes of groups have been defined by their relation to the cyclic groups:

===Virtually cyclic groups===
A group is called '''virtually cyclic''' if it contains a cyclic subgroup of finite [[index (group theory)|index]] (the number of [[coset]]s that the subgroup has). In other words, any element in a virtually cyclic group can be arrived at by multiplying a member of the cyclic subgroup and a member of a certain finite set. Every cyclic group is virtually cyclic, as is every finite group. An infinite group is virtually cyclic if and only if it is [[finitely generated group|finitely generated]] and has exactly two [[End (graph theory)|ends]];{{refn|group=note|If ''G'' has two ends, the explicit structure of ''G'' is well known: ''G'' is an extension of a finite group by either the infinite cyclic group or the infinite dihedral group.<ref>{{Harv|Stallings|1970|pp=124–128}}. See in particular {{Google books|3Lyvsc694T4C|Groups of cohomological dimension one|page=126}}.</ref>}} an example of such a group is the [[direct product of groups|direct product]] of '''Z'''/''n'''''Z''' and '''Z''', in which the factor '''Z''' has finite index&nbsp;''n''. Every abelian subgroup of a [[hyperbolic group|Gromov hyperbolic group]] is virtually cyclic.<ref>{{Harv|Alonso|1991|loc = Corollary&nbsp;3.6}}.</ref>

===Procyclic groups===

A [[profinite group]] is called '''procyclic''' if it can be topologically generated by a single element. Examples of profinite groups include the profinite integers <math>\widehat{\Z}</math> or the ''p''-adic integers <math>\Z_p</math> for a prime number ''p''.

===Locally cyclic groups===
{{main|Locally cyclic group}}
A [[locally cyclic group]] is a group in which each [[finitely generated group|finitely generated]] subgroup is cyclic. An example is the additive group of the [[rational number]]s: every finite set of rational numbers is a set of integer multiples of a single [[unit fraction]], the inverse of their [[lowest common denominator]], and generates as a subgroup a cyclic group of integer multiples of this unit fraction. A group is locally cyclic if and only if its [[lattice of subgroups]] is a [[distributive lattice]].<ref>{{Harv|Ore|1938|pp = 247–269}}.</ref>

===Cyclically ordered groups===
{{main|Cyclically ordered group}}
A [[cyclically ordered group]] is a group together with a [[cyclic order]] preserved by the group structure. Every cyclic group can be given a structure as a cyclically ordered group, consistent with the ordering of the integers (or the integers modulo the order of the group).
Every finite subgroup of a cyclically ordered group is cyclic.<ref>{{Harv|Fuchs|2011|p=63|}}.</ref>

===Metacyclic and polycyclic groups===
A [[metacyclic group]] is a group containing a cyclic [[normal subgroup]] whose quotient is also cyclic.<ref>{{springer|id=M/m063550|title=Metacyclic group|author=A. L. Shmel'kin}}</ref> These groups include the cyclic groups, the [[dicyclic group]]s, and the [[direct product]]s of two cyclic groups. The [[polycyclic group]]s generalize metacyclic groups by allowing more than one level of [[group extension]]. A group is polycyclic if it has a finite descending sequence of subgroups, each of which is normal in the previous subgroup with a cyclic quotient, ending in the trivial group. Every finitely generated [[abelian group]] or [[nilpotent group]] is polycyclic.<ref>{{SpringerEOM|id=p/p073560|title=Polycyclic group}}</ref>

==See also==
*[[Cycle graph (group)]]
*[[Cyclic module]]
*[[Cyclic sieving]]
*[[Prüfer group]] ([[Countable set|countably infinite]] analogue)
*[[Circle group]] ([[Uncountable set|uncountably infinite]] analogue)

==Footnotes==

===Notes===
{{reflist|group=note}}

===Citations===
{{reflist|30em}}

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==Further reading==
*{{Citation | last1=Herstein | first1=I. N. | title=Abstract algebra | publisher=[[Prentice Hall]] | edition=3rd | isbn=978-0-13-374562-7 | mr=1375019  | year=1996 | pages=53–60}}

==External links==
*Milne, Group theory, http://www.jmilne.org/math/CourseNotes/gt.html
* [http://members.tripod.com/~dogschool/cyclic.html An introduction to cyclic groups]
*{{MathWorld|title=Cyclic Group|urlname=CyclicGroup}}
* [http://groupnames.org/#?cyclic Cyclic groups of small order on GroupNames]
* [https://onlinedegreemath.com/every-cyclic-group-is-abelian/ Every cyclic group is abelian]

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