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Attempted Proof:

Let $S = \{k: n^k-1 \equiv 0 \pmod{p}, k\in \mathbb{N}\}$. Minimal element $r$ exists by Well-Ordering. we show $r=q$. We note, $\gcd(p,n)=1$, otherwise $p \mid n^q-1 \implies p \mid 1$.

We have:

$n \not\equiv 1 \pmod{p}\implies n^2 \not\equiv n \pmod{p}$ ; [since $p , n$ are coprime and $p$ prime ]. Now, for any $0<j< i < r,$ $n^i \equiv n^j \pmod{p} \implies n^j(n^{i-j}-1) \equiv 0 \pmod{p} \implies (n^{i-j}-1) \equiv 0 \pmod{p} $, contradicting the minimality of $r$.

Now, we proceed to show that, all elements of $S$ are integral multiples of $r$. (Claim 1)

We have, $n^{tr} \equiv 1 \pmod{p} \implies n^{tr+l} \equiv n^l \pmod{p} $, for $0\leq l <r$. Now, for any two such $l$, say $c$ and $d$, with $d<c$ , $n^{tr+c} \equiv n^{tr+d} \pmod{p} \implies n^{tr+d}(n^{c-d}-1) \equiv 0 \pmod{p}$, which again contradicts our assumption that $r$ is the least element of $S$. Hence, Claim 1 is established. But, it implies that $q$ is composite (an impossibility); hence $r$ must be equal to $q$ .

Finally, by Fermat's Little Theorem, we have $n^{p-1} \equiv 1 \pmod{p}$ [ it holds, since $\gcd (p, n) =1$].

Therefore, $p-1$ must be an integral multiple of $q$, i.e. $q \mid p-1$.

PS: I am aware of the duplicates, but I want my solution checked. Thank you.

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You appear to be reinventing the wheel (here $\rm ord$ = order of an element). Using this it's a one-liner

$\!\bmod p\!:\ n^{\large q}\equiv 1\,\Rightarrow\, {\rm ord}\, n\mid q\,$ prime, so $\, {\rm ord}\,n = q\,$ by $\,n\not\equiv 1$, so $\ n^{\large p-1}\equiv 1\,$ implies $\, q\mid p\!-\!1$

Said more structurally your set $S$ is a nontrivial set of integers closed under subtraction, so a one-line Euclidean descent proof shows the its least positive element $\,r := {\rm ord}\, n\,$ divides every element of $S$.

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  • $\begingroup$ Order of the element in which Group? $U_n$? $\endgroup$ – Subhasis Biswas Feb 9 at 19:22
  • $\begingroup$ @Subhasis In the cyclic group generated by $n$ (note $\,n^{p-1}\equiv 1\,$ by Fermat). But you don't need to know any group theory to understand these fundamental properties of $\,\rm ord$ $\endgroup$ – Bill Dubuque Feb 9 at 19:28
  • $\begingroup$ @Subhasis I added a link to full proofs of these basic properties of $\rm ord.\ \ $ $\endgroup$ – Bill Dubuque Feb 9 at 19:33
  • $\begingroup$ @BillDubuque: I'd be interested in your thoughts on this question $\endgroup$ – J. W. Tanner Jul 1 at 20:23
  • $\begingroup$ @J.W.Tanner I added some links there. If you chase them you should fine much of interest (including Moonshine!). $\endgroup$ – Bill Dubuque Jul 1 at 20:48

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