Show that in $\Bbb C^∗$ the subgroup $H = <i>$ is isomorphic to $\Bbb Z_4$.
So I know I need to show that $H$ is homomorphic to $\Bbb Z_4$ and then that there is a bijection, $f: H \to \Bbb Z_4$. I need some help getting started with this.
To begin, I think $H$ is cyclic. So listing everything out, $H = \{1, i, -1, -i\}$. I don't really know where to go from here. I think that $|H|=|\Bbb Z_4| =4$ So if $\Bbb Z_4$ is cyclic then that would mean that they are isomorphic. I'm not sure if it is indeed cyclic though. There are probably multiple ways to do this but this is the only thing coming to mind at the moment. Any help would be appreciated.