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.

(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 \[{e^{\frac{{2\pi i}}{n}}}\]