Editing Cyclic group (section)

===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>