In functional analysis, there are many examples of things that ``go wrong'' in the nonseparable setting. For instance, my favorite version of the spectral theorem only works for operators on a separable Hilbert space. Within C${}^*$-algebra there are many examples of nice results that require separability. (Dixmier's problem: is every prime C${}^*$-algebra primitive? Yes for separable C*-algebras, no in general.)

I wondered whether there is a similar phenomenon in pure algebra. Are there good examples of results that hold for countable groups, countable dimensional vector spaces, etc., but fail in general?

One example I know about is Whitehead's problem, which has a positive solution for countable abelian groups, but is independent of ZFC in general.

In functional analysis, there are many examples of things that "go wrong" in the nonseparable setting. For instance, my favorite version of the spectral theorem only works for operators on a separable Hilbert space. Within C${}^*$-algebra there are many examples of nice results that require separability. (Dixmier's problem: is every prime C${}^*$-algebra primitive? Yes for separable C*-algebras, no in general.)

I wondered whether there is a similar phenomenon in pure algebra. Are there good examples of results that hold for countable groups, countable dimensional vector spaces, etc., but fail in general?

One example I know about is Whitehead's problem, which has a positive solution for countable abelian groups, but is independent of ZFC in general.