Problem about primitive root

Question

Let p be a prime such that $q = \frac{p − 1}{2}$ is also prime. Suppose that g is an integer satisfying

$g \not\equiv \pm 1 \pmod p$ and $g^q \not\equiv 1 \pmod p$.

Prove that g is a primitive root modulo p.

Answer 1

Hint: The order of $g$ divides $p-1$. But since $(p-1)/2$ is prime, $p-1$ has very few divisors! And we are told that $g^q\not\equiv 1\pmod{p}$, which rules out one of them as the order of $g$. You can decide why $2$ and $1$ are ruled out.