QUESTION: does there exist a group U such that three conditions hold:
- (a) every finitely generated group is isomorphic to a subgroup of U;
- (b) for every group G that is not finitely generated there exists a group H that is generated by G and one additional element x, and such that H is not isomorphic to any subgroup of U;
- (c) (EDIT The following condition should not be here, please, forget it -- similar conditions would make sense for non-algebraic categories) U is generated by a countable set.
In the middle sixties (of the previous millennium) I posed a similar problem (almost a theorem :-) for compact metric spaces (instead of the finitely generated groups). In general, one can formulate similar problems for many different categories.
*Remark: The question seems to make sense even in the case of finite group; this time even the third condition too is reasonable to consider.