Simple group question need help...

Question

Alright so I've got a question here in terms of groups.

So define $\omega = {e}^{2i\pi\over 13}$ -The exponent of e should be $2i\pi\over 13$ but it's not coming clear when as an exponent of e there for some reason.

And then define the group G={$\phi_k:1\leq k \leq 12$}, where $\phi_k(\omega) = \omega^{k}$.

Then it says $\phi_2$ is easily checked to have order 12. I don't know why this is yet sounds very simple. Can anyone explain? Thank You.

Answer 1

Here $\phi_2=w ^ 2=e^{4i\pi/13}$ and I think it would be of order 13 since:

$\phi_2^{13}=e^{4i\pi}=1$

OR

It should be $ w=e^{2i\pi/12 }$ then $\phi_2 $ would be of order 12