Cyclic

Formal Definition

A group is cyclic if there is some element $a$ in $G$ that generates $G$.

Informal Definition

A cyclic group is a group that can be generated by a single element.

Example(s)

The group $\mathbb Z$ under addition is a cyclic group.

Non-example(s)

The group $\mathbb{R}$ under addition is not a cyclic group.

Let $G$ is a group with $a \in G$. Then the subgroup $\{ a^n \mid n \in \mathbb{Z} \}$ of $G$, characterized in Theorem 5.17, is called the cyclic subgroup of $G$ generated by $a$, and is denoted by $\langle a \rangle$.