Cyclic group

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.

Explanations

L1 human

A cyclic group is a group in which each element is a power of one fixed element. The rotations of a regular polygon with $n$ sides form a cyclic group: each rotation is a power of the smallest rotation. The word "cyclic" comes from this picture of going around a circle.