Since all surjective functions are epic, it follows that surjective homomorphisms are too. The converse requires more care. Let be an epimorphism and . It is sufficient to show . Consider the set of left-cosets of in , and let be a unique symbol, , and denote the permutation group of the set . We define a homomorphism so that for any

Let be the permutation which swaps and and leaves everything else invariant, and let be its induced inner automorphism. Then is also a homomorphism. Now, for any it follows leaves both and fixed (the latter is by construction), and hence commutes with and thus . Hence , and since is epic . But thence follows that commutes with for all , wherefore must leave fixed, and thus and thus for all .