TITLE: Is it a subgroup?!
QUESTION [5 upvotes]: Let $G$ be a finite group,  $A$ an its subset and put $A^{-1}=\{ a^{-1}:a\in A\}$.
Is it true that if $A$ is symmetric (i.e., $A=A^{-1}$),  $G=AB$ and $|AB|=|A||B|$, for some $B\subseteq G$, then $A$ is a subgroup of $G$?
Note: The converse is true, i.e., if $A$ is a subgroup then there exists such a subset $B$.

REPLY [2 votes]: Another simple counter-example is: $G = \mathbb{Z}/6\mathbb{Z}$, $A = \{\overline{0}_6, \overline{1}_6, \overline{5}_6\}$, $B = \{\overline{0}_6, \overline{3}_6\}$.