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}}}