# $AB = BA$ in a group $G$

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 at 18:32
no, I've edited –  WLOG Feb 23 at 18:33

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 $G$.