Let $G = Z^*_p$ under multiplication, with $p$ being a prime $> 3$ and $|G|=p-1$ and $G$ is cyclic. $H = \{a^2\mid a \in G\}$. Want to prove $H < G$ and $|H| = (p-1)/2$.
I can show that $H < G$, but how do I find the order of $H$ if I have to use the natural group homomorphism?
I know that $|H|$ divides $p-1$.
Thank you for your help!