Question 43
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 $AA \neq A$, $|AA| = |A|$ and $AA \lt G$.
(The notation $H \lt G$ means that $H$ is a proper subgroup of the group $G$, i.e., a subgroup of $G$ different from $G$ itself.)
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 $AA \neq A$, $|AA| = |A|$ and $AA \lt G$.
(The notation $H \lt G$ means that $H$ is a proper subgroup of the group $G$, i.e., a subgroup of $G$ different from $G$ itself.)