Let $p,q$ be primes and $n \in \Bbb N,$ such that $p \nmid\ n-1.$ If $p\ |\ n^q-1$ then which one of the following option/s is/are correct?
$(1)$ $p\ |\ q-1.$
$(2)$ $q\ |\ p-1.$
$(3)$ $p\ |\ (p-1)(q-1).$
$(4)$ $q\ |\ (q-1)(p-1).$
My attempt $:$ If I take $p=3, q=2$ and $n=2$ then I find that the given condition holds but option $(1)$ and hence option $(3)$ fails to hold. What about options $(2)$ and $(4)$? It is quite clear that either both the options $(2)$ and $(4)$ are correct or both the options are false. How to prove or disprove $(2)$? Any help in this regard will be highly appreciated.
Thank you very much for your valuable time.