clarifying definition of undecidable Because there are examples of both abelian and non abelian groups. Does that mean a group being abelian is undecidable from the group axioms?
And, if so, what contexts and connotations does using the word undecidable introduce? It seems like I only ever see the term in logic and foundations. Why?
 A: 
Because there are examples of both abelian and non abelian groups. Does that mean a group being abelian is undecidable from the group axioms?

Yes. Precisely:


*

*$\varphi$ is provable from $T$ if, well, $T\vdash\varphi$; equivalently (by the Completeness Theorem), if every model of $T$ satisfies $\varphi$.

*$\varphi$ is disprovable from $T$ if $\neg\varphi$ is provable from $T$.

*$\varphi$ is unprovable from $T$ if $\varphi$ isn't provable from $T$, and $\varphi$ is undecidable from $T$ if neither $\varphi$ nor $\neg\varphi$ is provable from $T$.

And, if so, what contexts and connotations does using the word undecidable introduce? It seems like I only ever see the term in logic and foundations. Why?

Undecidability emerges as a property of interest precisely when we're looking at theories that might be complete. E.g. it can be surprising to learn that PA isn't complete, that is, that there are sentences in the language of arithmetic which are undecidable from PA. By contrast, there was never a thought that the group axioms would be complete, precisely because they emerged to describe a class of wildly variable structures (namely, groups). So while you can say that commutativity is undecidable in the theory of groups, it sounds a bit off.
Another important point is that the attention given to a theory as a syntactic object, rather than just the class of objects it corresponds to, is a key piece of logic: most of the time in group theory, one doesn't really ever need to use the theory of groups, we just reason about groups. So the theory of groups itself isn't really an object that crops up frequently and explicitly unless logic is on the mind somewhere.$^*$

$^*$Counterexamples welcome!
