Editing Cyclic group (section)

==Examples==
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===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>