# Are abelian subgroups normal?

Let $G$ be a group, and let $H < G$ be such that all the elements of $H$ commute with each other, i.e. $H$ is abelian. Then is $H$ necessarily normal in $G$, i.e. $H \unlhd G$?

No. One counterexample is the subgroup $H=\langle (12)\rangle$ of $G=S_3$.
No. Take $G=S_3$ and $H=\langle (1,2)\rangle$.