While revising I came across this question:
Let $G$ be the group of complex numbers $\{1, i, -1, -i\}$ under multiplication.
Is it true that for every such homomorphism, there is an integer $k$ such that the homomorphism has the form $z\mapsto z^k$ ?
The answer is true, but how does one prove it?
I thought of considering four cases of $f(i)=1,i,-1,-i$, since $i$ is a generator.
Sincere thanks for any help.