Subgroup Test (one-step)

Subgroup Test (one-step): Let $G$ be a group and let $H$ be a nonempty subset of $G$. Then $H$ is a subgroup of $G$ if and only if for all $x,y\in H$ one has $xy^{-1}\in H$.

This is often the fastest criterion to check the subgroup property because it packages “closed under products” and “closed under inverses” into a single closure condition.