If $G$ is a group and $H_1,H_2$ are cyclic subgroups of $G$ with $G=H_1H_2=H_2H_1$ then must $G$ be cyclic?
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Note that if $H_1$ and $H_2$ are subgroups of $G$, then $H_1H_2$ is a subgroup if and only if $H_1H_2 = H_2H_1$. So the assumption $G = H_1H_2 = H_2H_1$ is equivalent to $G = H_1H_2$.
One special case where $G = HK$ is when $G$ is the direct product of $H$ and $K$.
Is the direct product of two cyclic groups always cyclic?