Let be a subgroup of order . Since is normal subgroup of we can form a factor group . The order of this factor group is . Let . And consider the coset . Lagrange's theorem tells us that divides . But since and it means (again by Lagrange's theorem). Hence, . Thus, divides (by the property of orders). But since it means is forced to be . Thus, and so . This means . And this completes the proof (that i.e. it is unique) because if it were the case that then it would mean which is impossible.