Suppose that G, G', and G'' are groups. If G' is a homomorphic image of G, and G'' is a homomorphic image of G', prove that G'' is a homomorphic image of G.
Something tells me that it involve transitive property ...
Obviously.
So, we have which are both epimorphisms (surjective homomorphisms). So the question is why is an epimorphism? But this is easy when put this way since the product of surjective maps is surjective and the product of homomoprhisms is a homomorphism. Prove those facts if you don't know them.