Two left cosets are equal exactly when their representatives differ by an element of the subgroup.Suggest a slogan
Let \(G\) be a group, \(H\leq G\) a subgroup, and \(a,b\in G\). Then \[ aH = bH \iff a^{-1} b \in H \iff b^{-1} a\in H. \]
If \(aH=bH\), then \(b=b\cdot e\in aH\), so \(b=ah\) with \(h\in H\) and \(a^{-1}b=h\in H\). If \(a^{-1}b=h\in H\), then \(bH=ahH=aH\). Finally, \(b^{-1}a=(a^{-1}b)^{-1}\), and \(H\) is closed under inverses.