Integers modulo n form a cyclic group

Integers modulo n form a cyclic group
Tag	`FB79N7AHZA`
URI	`m2://FB79N7AHZA`
Kind	example
Writing	W0 stub
MSC	20A05
Instance of	Cyclic group, Abelian group
Illustrates	Cyclic groups are abelian
Uses	Integers modulo n, Cyclic group

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Example

Let $n \ge 1$ . The integers modulo $n$ , $\mathbb{Z}/n\mathbb{Z} = \{0, 1, \dots, n-1\}$ , form a group under addition modulo $n$ . Each element $k$ is the sum $1 + 1 + \dots + 1$ of $k$ copies of $1$ . Thus $1$ is a generator, and the group is cyclic of order $n$ . Addition modulo $n$ is commutative, so the group is abelian. This agrees with the result that cyclic groups are abelian.

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Group concept
Integers modulo n concept
Laws of exponents in a group statement
Subgroup definition
Cyclic subgroup generated by an element definition
Cyclic group definition
Integers modulo n form a cyclic group example

Sources

Web https://en.wikipedia.org/wiki/Cyclic_group, Integer and modular addition

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