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Well, I know of some examples of groups which are trivial enough and of some which maybe are not so trivial.

It could be the case that we could construct some operation on some set which is such that it might be hard to determine is that set with that operation a group, but I am interested here in some examples which are of some importance in mathematics.

So the question would be:

Are there any examples in mathematics of sets (equipped with some operation) for which it is not known are they groups and which are such that the determination of the question are they groups or not would settle some open problem/conjecture?

Farewell
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There are no such examples. Importance of the group theory (in the modern mathematics) comes, primarily, from two sources:

  1. Groups provide some interesting invariants of the object one is studying. For instance, groups in algebraic topology (homology, cohomology and homotopy groups).

  2. Existence of a group structure or existence a group acting as a symmetry group, provides an extra insight into a problem one is trying to analyze. Examples are Galois theory (in algebra), symmetry groups of differential equations (e.g. where large symmetry group might lead to complete integrability, also in the gauge theory), number theory (group structure of the set of rational points of an elliptic curve, for instance, take a look here: https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture).

In most cases, the fact that one does have a group structure is either immediate or almost immediate. See, however, answers to this question.

Community
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Moishe Kohan
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    Just because you haven't encountered any does not mean they do not exist. I would not be surprised if there was a nice example. At the very least you could construct artificial examples based on well-known conjectures. e.g., take $G = { -2 + \liminf_{n\rightarrow \infty} p_{n+1} - p_n}$ under addition where $p_n$ is the $n$-th prime. The Twin Prime conjecture says this is the group ${0}$. – Jair Taylor Apr 19 '16 at 18:46
  • @JairTaylor: Yes, one can construct artificial examples. As for your example, the open problem is not whether it is a group but what it is as a set. – Moishe Kohan Apr 19 '16 at 21:24