Are abelian subgroups normal?
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 $.
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 $.