Suppose that f: G → H is a homomorphism of groups and also a surjection. In Chapter 17, you saw that the kernel of a homomorphism consisted of

those elements which map onto the identity element, and that the solution for x ∈ G of an equation such as f(x) = a is a coset in G. You find one element g ∈ G such that f(g) = a, and then you find the coset gK, where K = ker f.