[Math] Can a non-abelian subgroup be such that the right cosets equal the left cosets

Question

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?

Answer 1

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 $8$.