@Maurice: Interesting! -- Anyway I'd be interested to see some concrete examples. To start with something (supposedly!) easy, would it be feasible to find a finite presentation for a group whose orders of torsion elements are precisely the primes congruent to 1 mod 4? -- Knowing that something exists is one important thing, but explicitly finding it is still more (in particular for someone like me with a background in explicit computation!).

@Maurice: I wonder to what extent these nice results can be turned into practical algorithms. Thus for example for which sets $X$ of integers it would be computationally feasible to obtain a finite presentation of a group which has precisely the $n \in X$ as orders of torsion elements. Also it would be interesting to see how long the resulting presentations would be in concrete nontrivial examples.

In the comment it does compile -- strange.

Another formatting issue: the expression $F_{\infty}*{a \in A}(*{j \in \pi(a)}C_{j})$ does not seem to compile.

I posted this question since I wondered how rich the class of finitely presented groups is in a certain sense. I would have found finitely presented groups rather boring if the answer to my question would have been "yes". -- But now the answers tell me that the class of finitely presented groups is richer than I thought!

Further minor remarks: 1. I think "$1 \Rightarrow 2" and "$2 \Rightarrow 1" at the beginning of your post need to be interchanged(?) 2. "factors" should rather mean "divisors", right?

Minor remark: on this site, for some reasons set brackets need to be escaped by two backslashes -- otherwise they vanish.

Wow, still a more general result ... -- Great! -- Thanks!

Right. -- So now $f_p(n)$ is at least $O(n \cdot \ln{n})$ and at most $O(p^{n-1})$. Still a wide range.

Right. -- Now the growth can still be anything from linear to exponential.