Cyclic Group

A group $G$ is called a cyclic group if, for some $a \in G$, every element $x \in G$ is of the form ${a^n}$, where $n$ is some integer. The element $a$ is called $a$ generator of $G$.

There may be more than one generator of a cyclic group. If $G$ is a cyclic group generated by $a$, then we shall write $G = \left\{ a \right\}$ or $G = < a >$. The elements of $G$ will be of the form $\ldots ,{a^{ - 3}},{a^{ - 2}},{a^{ - 1}},{a^0},{a^1},{a^2},{a^3}, \ldots$. Of course, they are not necessarily all distinct.

Examples:
(1) The multiplicative group $\left\{ {1,\omega ,{\omega ^2}} \right\}$ is cyclic. The generators are $\omega$ and ${\omega ^2}$.

(2) The multiplicative group $G = \left\{ {1, - 1,i, - i} \right\}$ is cyclic. We can write $G = \left\{ {i,{i^2},{i^3},{i^4}} \right\}$. The generators are $i$ and $- i$.

(3) The multiplicative group of $n$, nth roots of unity are cyclic, a generator being

$${e^{\frac{{2\pi i}}{n}}}$$