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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. $H \unlhd G$?

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No. One counterexample is the subgroup $H=\langle (12)\rangle$ of $G=S_3$.

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No. Take $G=S_3$ and $H=\langle (1,2)\rangle $.

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