Cyclic Group

A group G is called 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 that called a generator of G.

There may be more than one generators 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 generator 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 is cyclic, a generator being

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