Suppose G is a group and for every $H_1,H_2\le G$, $H_1H_2=H_2H_1$. Is $G$ abelian?
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Let $H_1$ and $H_2$ be subgroups of $G$. Then $H_1H_2 = H_2H_1$ if and only if $H_1H_2$ is a subgroup.
Thus if for example every subgroup of $G$ is normal, then $H_1H_2$ is always a subgroup. The quaternion group of order $8$ is a nonabelian group where every subgroup is normal.