Does there exist any uncountable group , every proper subgroup of which is countable? Does there exist an uncountable group , every proper subgroup of which is  countable ? 
 A: The existence of such groups is apparently proved in the paper Shelah, Saharon "On a problem of Kurosh, Jónsson groups, and applications". Word problems, II (Conf. on Decision Problems in Algebra, Oxford, 1976), pp. 373–394, Stud. Logic Foundations Math., 95, North-Holland, Amsterdam-New York, 1980.
I haven't seen the paper itself, but here is the description in Math. Reviews from MathSciNet.
Using small cancellation theory, the author constructs some remarkable infinite groups, notably "Jónsson groups'', that is, groups of uncountable cardinality $\lambda$ containing no proper subgroups (or better: no proper subsemigroup) of the same cardinality. One construction works for all successor cardinals if we assume the generalized continuum hypothesis; a variant works for $\aleph_1$ with no set-theoretic assumptions.
The groups constructed actually have slightly stronger properties and hence constitute counterexamples to a few not obviously related conjectures, notably (Theorem C): The continuum hypothesis implies that there is an uncountable group which admits no nontrivial topology (i.e. cannot be made into a topological group in a nontrivial way). 
