Subgroup test

A nonempty subset $H$ of a group $G$ is a subgroup if and only if $g{h}^{-1}\in H$ for every $g,h\in H$. Necessity follows from closure and the existence of inverse elements in a subgroup.

Conversely, choose $a\in H$. Setting $g=h=a$ gives the identity element $e=a{a}^{-1}\in H$. Taking $g=e$ then gives $h^{-1}\in H$ for every $h\in H$. Finally, using $h^{-1}$ as the second element gives $g(h^{-1}{)}^{-1}=gh\in H$, proving closure. The inherited multiplication is associative, so these properties establish the group axioms on $H$.