# Definition:Cyclic Group/Generator

## Definition

Let $G$ be a cyclic group generated by the element $g$.

Let $a \in G$ be an element of $G$ such that $\gen a = G$.

Then $a$ is a generator of $G$.

## Examples

### Subgroup of $\struct {\R_{\ne 0}, \times}$ Generated by $2$

Consider the multiplicative group of real numbers $\struct {\R_{\ne 0}, \times}$.

Consider the subgroup $\gen 2$ of $\struct {\R_{\ne 0}, \times}$ generated by $2$.

Then:

$\dfrac 1 2$ is also a generator of $\gen 2$

but:

$4$ is not a generator of $\gen 2$.

### Generators of $C_8$

Let $C_8$ be generated by $x$:

$C_8 = \gen x$

The set of generators of the cyclic group $C_8$ is:

$\set {x, x^3, x^5, x^7}$