Group theory and generalizations Definition

Cyclic group

Builds on Cyclic subgroup generated by an element

Definition

A group $G$ is cyclic when there is an element $g \in G$ with $G = \langle g \rangle$ , where $\langle g \rangle$ is the cyclic subgroup generated by $g$ . Such an element $g$ is called a generator of $G$ .

Remarks

The group $(\mathbb{Z}, +)$ is cyclic, and its generators are $1$ and $-1$ . The integers modulo $n$ under addition form a cyclic group of order $n$ , generated by $1$ . A cyclic group can have many generators: in the integers modulo $n$ , the generators are the residues coprime to $n$ . The circle group is not cyclic, because it is uncountable.

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

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Tag	`0B85Q41RP6`
URI	`m2://0B85Q41RP6`
Kind	definition
Writing	W2 usable
MSC	20A05
Defines	Cyclic group
Uses	Cyclic subgroup generated by an element

Cyclic group

Definition

Remarks

Reading path

Where this leads

Further reading

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