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Suppose $G$ is a group , $ A \subset G $ and $ B \subset G $ are subsets of $G$, if $AB = BA$ is it true that $AB$ is a subgroup of $G$ ? Why ?

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Do you assume that $A$ and $B$ are subgroups of $G$? – Singhal Feb 23 '14 at 18:32
no, I've edited – WLOG Feb 23 '14 at 18:33
up vote 7 down vote accepted

If $A$ and $B$ are not assumed to be subgroups of $G$ themselves, then the answer is trivial. Take $G = \Bbb{Z}$ and let $A = B = \{ 1 \}$. Then, $AB = BA = \{ 2 \}$. But, clearly $AB$ is not a subgroup of $G$.

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