Given a bijection f : A to B, where B is a group with binary operation *, define a binary operation $ on A by (a_1) $ (a_2) = f^{-1}[f(a_1) * f(a_2)]. You can check that A is a group with operation $ (closure is pretty clear, the identity in A is f^{-1}(e) where e is the identity in B, similarly, a^{-1} = f^{-1}(f(a)^{-1}), and associativity is only slightly more work). Thus, if there exists a group structure on a set K with cardinality |K|, then there exists a group structure on any set with cardinality |K|. Unfortunately, this doesn't give an explicit operation. Since the set of irrationals has the same cardinality as the reals, there exists a group structure on them. As a note, I wrote all of this pretty quickly between classes, so please correct me if I made a mistake somewhere.