Show that group $G$ is isomorphic to $Z_3$.

Question

I have this question:

Let $w = e^{\frac{2 \pi i}{3}}$, that is $w^3 = 1$ and $G = {w^0, w^1, w^2}$ or $G = {1,\frac{\frac{-1}{2} + i \sqrt{3}}{2},\frac{\frac{-1}{2} - i \sqrt{3}}{2}}$. Show that $G$ is $\mathbb{Z}_3$ isomorphic.

I set up Cayley's table for the two groups, but I can't see or justify why the two are isomorphic.

I saw that the two groups maintain the same cyclicality, but is that enough to say that they are isomorphic?

Thanks.

Answer 1

Hint: The elements $w^0, w^1, w^2$ correspond to the vertices of a triangle in the complex plane. Multiplication by $w^1$ permutes these vertices cyclically.

Addition of $[1]_3$ in $\Bbb Z_3$ permutes the elements of $\Bbb Z_3$ cyclically too.