Let $(G, \cdot)$ be a group with identity element $e$. A subset $H \subset G$ is a subgroup of $G$, written $H \le G$, if the following three conditions hold:
1. Identity: $e \in H$.
2. Closure: for all $h_1, h_2 \in H$, $h_1 \cdot h_2 \in H$.
3. Inverses: for all $h \in H$, $h^{-1} \in H$.
A subgroup $H$ of $G$ is proper if $H \neq G$, written $H < G$.