Editing Cyclic group (section)

==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>}}
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