Simple/natural questions in group theory whose answers depend on set theory

Solution 1:

One of my favourite results, told to me by my advisor over coffee once.

Let $G$ be an abelian group. We say that it has a norm if there is a function $\nu\colon G\to\Bbb R$ whose behaviour is what you'd expect from "norm".

Say that a norm is discrete if its range in $\Bbb R$ is a discrete set.

Exercise. If $G$ is a free-abelian group, then it has a discrete norm.

Difficult theorem. If $G$ has a discrete norm, then it is free-abelian.

The only known proof uses Shelah's compactness theorem for singular cardinals. So quite significant heavy machinery from set theory and model theory combined.

Solution 2:

Turning my comment into an answer. Given a group $G$ we can define its dual group $G^\ast=\mathrm{Hom}(G,\Bbb Z)$ and, just like for vector spaces, we get a canonical "evaluation" homomorphism $G\to G^{\ast\ast}$ given by $g\mapsto(f\mapsto f(g))$ and we call reflexive the groups for which this homomorphism is an isomorphism.

Theorem: Every free abelian group is reflexive iff there is no measurable cardinal.

I'll prove the $\implies$ direction, the other one is not as easy. Let $\kappa$ be measurable, and let $\mathcal U$ a $\kappa$-complete nonprincipal ultrafilter on $\kappa$.
Consider the free Abelian group $\Bbb Z^{(\kappa)}=\bigoplus_{i<\kappa}\Bbb Z$ with its standard basis $\{e_\xi\}_{\xi<\kappa}$ and note that $\mathrm{Hom}(\Bbb Z^{(\kappa)},\Bbb Z)\simeq\Bbb Z^\kappa$.
Consider the function $\varphi\colon\Bbb Z^\kappa\to\Bbb Z$ given by $\varphi(x)=n$ iff $\{\xi\in\kappa\mid x(\xi)=n\}\in \mathcal U$, we claim that this function is not in the image of the canonical homomorphism $j\colon\Bbb Z^{(\kappa)}\to(\Bbb Z^{(\kappa)})^{\ast\ast}\simeq (\Bbb Z^\kappa)^\ast$, indeed for every nonzero $x\in\Bbb Z^{(\kappa)}$ we have $j(x)(e_\xi)=x(\xi)\neq 0$ for some $\xi$, but $\varphi(e_\xi)=0$ for every $\xi$. (in the identification $(\Bbb Z^{(\kappa)})^\ast\simeq \Bbb Z^k$, $e_\xi$ corresponds to the projection $\Bbb Z^\kappa\to\Bbb Z$ on the $\xi$-th factor).

A reference for the proof of the other direction (actually of a much more general result which trivially implies the other direction) is Corollary III.3.8 in the beautiful book "Almost free modules: set theoretic methods", by Eklof and Mekler.

Solution 3:

There's the Whitehead problem : let $A$ be an abelian group such that any extension $0\to \mathbb Z\to K\to A\to 0$ (with $K$ abelian) splits.

Is $A$ then necessarily a free abelian group ?

The question is famously independent of ZFC, so depends on set theory.