Subgroup criterion

Statement

Let $G$ be a group and let $H$ be a subset of $G$. Then $H$ is a subgroup of $G$ if and only if $H$ is not empty and $ab^{-1} \in H$ for all $a, b \in H$.

Remarks

If $a \in H$, then $e = aa^{-1} \in H$, then $b^{-1} = eb^{-1} \in H$ for $b \in H$, and then $ab = a(b^{-1})^{-1} \in H$. In additive notation, the condition is $a - b \in H$ for all $a, b \in H$. For a finite nonempty subset $H$, it is sufficient that $ab \in H$ for all $a, b \in H$.

Explanations

L1 human

This test replaces three checks by one. You only check that the subset is not empty and that $ab^{-1}$ stays in the subset. For example, the multiples of $5$ form a subgroup of $(\mathbb{Z}, +)$, because the difference of two multiples of $5$ is a multiple of $5$.