75th Romanian Mathematical Olympiad

Let $(G,\cdot )$ be a group, with the unit element $e$, and $A$ a non-empty subset of $G$. We denote $AA=\{xy\mid x,y\in A\}$.

a) Show that if $G$ is finite, then $AA=A$ if and only if $e\in A$ and $|AA|=|A|$.

b) Give an example of a group $G$ and a subset $A\subseteq G$, such that $AA\ne A$, $|AA|=|A|$ and $AA<G$.

(The notation $H<G$ means that $H$ is a proper subgroup of the group $G$, i.e., a subgroup of $G$ different from $G$ itself.)