A nonempty subset is a subgroup once it is closed under $ab^{-1}$.Suggest a slogan
Let \(G\) be a group and \(H \subseteq G\) a nonempty subset. If \(ab^{-1}\in H\) for all \(a,b\in H\), then \(H\) is a subgroup of \(G\).
Choose \(a\in H\); then \(1=aa^{-1}\in H\). For \(b\in H\), \(b^{-1}=1\cdot b^{-1}\in H\). For \(a,b\in H\), \(ab=a(b^{-1})^{-1}\in H\).