Subset Product of Abelian Subgroups
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Theorem
Let $\left({G, \circ}\right)$ be an abelian group.
Let $H_1$ and $H_2$ be subgroups of $G$.
Then $H_1 \circ H_2$ is a subgroup of $G$.
Proof
From Subgroup of Abelian Group is Normal, $H_1$ and $H_2$ are normal.
The result follows from Subset Product with Normal Subgroup is Subgroup‎.
$\blacksquare$