Question 66
Let $(G, \cdot)$ be a group with the unit element $e$, and $H$ and $K$ two proper subgroups of $G$, such that $H \cap K = \{e\}$ and the set $(G \setminus (H \cup K)) \cup \{e\}$ is closed with respect to the operation in $G$. Show that $x^2 = e$, for any $x \in G$.