The homomorphism p:G-->H induces an isomorphism between G/Ker(p) and H (if p is onto). I am trying to understand why this must be true. I understand why these groups have the same magnitude and so a bijection is possible, but there is something that I am not able to understand. What seems to be true but I don't understand why is that if a and b are elements in G, that if p(a) = p(b) then aK = bK in the factor group. Is this true? It seems like it should be. If so, then can somebody help me understand why it is true on an intuitive level? Sorry, I realize that this is a broad and poorly phrased question, hoping somebody can see through my lack of competent communication and give me some insight here.