Generator of Cyclic Group/Examples/C8

Examples of Generators of Cyclic Groups

Let $C_8$ be generated by $x$:

$C_8 = \gen x$

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

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

Proof

From Power of Generator of Cyclic Group is Generator iff Power is Coprime with Order, $C_8$ is generated by any $x^n$ where $n \perp 8$.

Hence the result.

$\blacksquare$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $5$: Subgroups: Exercise $14$