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:

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

Also see

• Results about generators of groups can be found here.

Sources

• 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $0$: Some Conventions and some Basic Facts
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 39$. Cyclic Groups