Subgroups are closed under inverses and products

Proposition (Closure properties of subgroups). Let $H$ be a subgroup of a group $G$. Then:

1. The identity element $e$ of $G$ lies in $H$.
2. If $h\in H$, then $h^{-1}\in H$.
3. If $h_1,h_2\in H$, then $h_1h_2\in H$.
4. Consequently, if $h_1,h_2\in H$, then $h_1h_2^{-1}\in H$.

Context. The reverse implication (a nonempty subset closed under $h_1h_2^{-1}$ is a subgroup) is packaged as a subgroup test , so this proposition is the “easy direction” used constantly in computations.