Let \(G\) be a group. Define a relation \(\sim\) on \(G\) such that \(a \sim b\) when \(\exists x \in G : a = xbx^{-1}\). Then \(\sim\) is an equivalence relation on \(G\).

Let \(G\) be a group and \(H \le G\). Define \(\equiv\) such that \(x \equiv y\) when \(xy^{-1} \in H\). Then \(\equiv\) is an equivalence relation on \(G\).