Is there a definition of a good problem in mathematics?

I'm working on a problem in $p$-group theory, namely "is it true that every finite non-simple $p$-group has a non-inner automorphism of order $p$?". (known to be posed or conjectured by Y. Berkovich)

My feeling about this question is being somewhat particular and isolated. I asked some specialists (who I'm really indebted to them) about this question, and their answers were very different: well known and quite difficult (a clever answer); important; particular and not very interesting; perhaps the methods that may be developed upon it, may have a great interest .

What make this isolated problem (with many many others) less interesting than Fermat's Last Theorem (conjecture) before its demonstration?

More generally, what make a problem in mathematics important or not?

Perhaps a problem is impotant if its solution implies the solution of other problems, but this amounts to saying that a set of problems is interesting, so we have the same problem.

And if we wish just to understand some things by solving a problem, then all the problems are interesting.

I'm sorry if the question is so vague.

• Good and important are orthogonal features (both are subjective, so this is not really true, but is a good enough approximation). Important problem is one which would have many implications if solved (e.g. $P = NP$ or the Riemann hypothesis). On the other hand there are good problems which does not have any profound meaning, but are intrinsically nice, some examples can be found in contest problems, or known puzzles (one of the reasons they became puzzles or contest problems is because their particularly beautiful solutions). – dtldarek Oct 21 '13 at 20:48
• Fermat's Last Theorem was interesting because of the mathematics surrounding it. Taken in isolation, it seems like an exotic problem without much interest. – Bruno Joyal Oct 22 '13 at 0:39