Show the klein group and <i> is not isomorphic

Question

One part in particular that I'm stuck on (other than the whole thing, so seeing the solution will help) is why is order 2? To me, it should be 4 as {1, i, -1, -i}.

Answer 1

$<i>$ is cyclic $i, i^2=-1, i^3=-1(i)=-i, i^4=1$ but not is the Klein group which is $\mathbb{Z}/2\times\mathbb{Z}/2$.

Answer 2

The order of $i$ is $4$, you are absolutely right.

On the other hand every element of the Klein four-group is of order $2$. Hence they cannot be isomorphic.