How we can prove that if $p,q$ be prime and $q\mid a^p-1$, then $q\mid a-1$ or $q=2kp+1$?

Question

As an elementary number theory question I tried to prove that if $p,q$ be prime and $q\mid a^p-1$, then $q\mid a-1$ or $q=2kp+1$?

I tried to solve this question but I stuck

Answer 1

Hint By Fermat Little Theorem $$ a^{q-1} \equiv 1 \pmod{q} $$ You also know that $$ a^p \equiv 1 \pmod{q} $$ Let $d =\mbox{gcd}(p,q-1)$. Show that $a^{d} \equiv 1 \pmod{q}$.

Finally $d|p$ and $p$ is prime, you only have 2 choices for $d$. Discuss both.