Group theory and generalizations Example

Cyclic group of order n

Example

Let $n$ be a positive integer. The integers modulo $n$ , $\mathbb{Z}/n\mathbb{Z} = \{0, 1, \ldots, n - 1\}$ , with addition modulo $n$ , form a cyclic group of order $n$ . The residue $1$ generates it, because each residue $k$ is the sum of $k$ copies of $1$ . Each cyclic group of order $n$ is isomorphic to $\mathbb{Z}/n\mathbb{Z}$ , and this group is often written $C_n$ .

Remarks

A residue $k$ generates $\mathbb{Z}/n\mathbb{Z}$ exactly when $\gcd(k, n) = 1$ . The group of $n$ th roots of unity is a cyclic group of order $n$ in multiplicative notation. The rotations of a regular $n$ -gon also form a cyclic group of order $n$ .

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Group concept
Laws of exponents in a group statement
Order of a group definition
Subgroup definition
Cyclic subgroup generated by an element definition
Cyclic group definition
Cyclic group of order n example

Tag	`0G0B3DRG65`
URI	`m2://0G0B3DRG65`
Aliases	`integers-modulo-n-under-addition`
Kind	example
Writing	W0 stub
MSC	20K01
Instance of	Cyclic group
Uses	Cyclic group, Order of a group

Cyclic group of order n

Example

Remarks

Reading path

Further reading

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