Let $\displaystyle A$ be a finite subset of a group $\displaystyle G$ (not necessarily a subgroup). Denote by $\displaystyle A^2$ the set $\displaystyle \{a_1 a_2 | a_1, a_2 \in A\}$.

Prove that $\displaystyle |A^2| = |A| \iff $ the following is true:

$\displaystyle A$ equals a left coset $\displaystyle aH$ for some subgroup $\displaystyle H \leq G$ and some

element $\displaystyle a \in G$, and $\displaystyle A$ also equals some right coset $\displaystyle Hb$, $\displaystyle b \in G$.

The backward implication is easy enough since all that's required is a simple bijection, but I'm having problems proving the forward implication.

Thanks in advance!