Let $\mathcal{C}$ be any specified set or class whose objects are groups. Define a relation $\sim$ on $\mathcal{C}$ by declaring that, for groups $G,H \in \mathcal{C}$, one has $G \sim H$ if and only if there exists a [group isomorphism](/page/Group%20Isomorphism) $\varphi: G \to H$. Then $\sim$ is an [equivalence relation](/page/Equivalence%20Relation) on $\mathcal{C}$.