A while ago I asked the question "Is there a group-theoretic proof that multiplicative groups of integers modulo a prime are cyclic?". Instead of an answer I got a few comments from competent users (including Keith Conrad, whose paper I mentioned in the question) telling me that the answer was, in short, 'no'.
Lately the question has been popping into my mind anew, and thus I thought about adding a bounty to the post, but I have a few concerns:
Is it OK to ask a question that may have no answer? I'm fine with a few of my reputation points going to waste if there is indeed no proof of the kind. I worry instead that making a question that may have no answer goes against what the site is intended to be.
Would it be discourteous (which I genuinely do not intend to be) to the users who left the comments? The comments are, I should mention, quite reasonable. As an example Conrad's comment reads
...the multiplicative group of integers modulo m is generally not cyclic, so proving it is when m is a prime number (or an odd prime power) will need to use something that distinguishes those choices of m from others, and a very basic one is that the multiplicative group of integers modulo a prime is a field, which is not a purely group-theoretic issue.
This is sensible, and the fact that whenever m is prime we get a field is a natural direction on which to build an idea as to why these groups are cyclic. I yet remain with a small hope that a group-theoretic proof could exist. Can I, with all being said, still add the bounty?