That's my question. I think it is, but can you proof this?
And is it both ways: and if a group has a abelian subgroup, is the group itself always abelian?
Yes, no.
Think about it! If a group is abelian, then $\displaystyle xy=yx \forall x,y \in G$, and so this definitely holds for any subset $\displaystyle S \subset G$.
On the other hand, if you take any group $\displaystyle G$, then the trivial subgroup $\displaystyle \{1\}$ is abelian, but $\displaystyle G$ isn't necessarily.