So, I know that every abelian (commutative) group $G$ is such that, for any subgroup $H$, the left cosets of $H$ in $G$ are the right cosets. I guess this is true even if $G$ is not abelian, but $H$ is (but I'm not sure). Is this enough to characterise the subgroup as abelian, or are there examples of non-abelian subgroups with this property?
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For a subgroup of a group, the condition of being abelian is neither necessary nor sufficient for being normal:
If $G$ is non-abelian, then $H=G$ is a normal but non-abelian subgroup. As another example, the alternating group is a normal but non-abelian subgroup of the symmetric group.
For a counterexample in the converse direction, consider the free group with two generators. The subgroup generated by one of the generators is abelian (isomorphic to $\mathbb Z$) but not normal. (This means that your guess above is wrong.)
The left cosets of $H$ are the same as the right cosets of $H$ if and only if $H$ is a normal subgroup of $G$. There are non-abelian groups all subgroups of which are normal, for example the quaternion group of order $8$.