Answer to Question #301136 in Abstract Algebra for rahul

Question

Let f : G → G0 be a group epimorphism, and let H be the normal subgroup that be the Kernal of the epimorphism. Then, prove that G0 is isomorphic to G/H.

Answer 1

By the universal property of a quotient, there is a natural homorphism f : G/H $\to$ G 0 .

Suppose that x is in the kernel of f. Then x has the form gH and by definition of f, f(x) = φ(g). Thus g is in the kernel of φ and so g ∈ H. In this case x = H, the identity of G/H. So the kernel of f is trivial and f is injective. Hence f is an isomorphism.