TITLE: Uncountable counterexamples in algebra
QUESTION [32 upvotes]: 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.

REPLY [9 votes]: Since you mentioned Whitehead's problem here is another interesting independence example.
For a group $G$ define its dual $G^\ast$ to be $\mathrm{Hom}(G,\Bbb Z)$. Like in the vector spaces case we get a canonical evaluation homomorphism $j\colon G\to G^{\ast\ast}$ given by $g\mapsto(f\mapsto f(g))$ and we call a group reflexive if $j$ is an isomorphism.
Now let $G$ be free abelian, must it be reflexive? The answer is provably positive for all "small" free abelian groups in $\mathsf{ZFC}$, in fact it is positive for all free abelian groups iff there is no measurable cardinal.