Three Theorems of Isomorphism
Three Theorems of Isomorphism: First Theorem of Isomorphism: If f: G →G' be an onto homomorphism with kernel K = ker f , then G/K ≈ G' In other words , every homomorphic image of a group G is isomorphic to a quotient group of G. Proof : Define a map Φ : G/K →G' such that , Φ (Ka) = f(a) , a∈G We Show Φ is an isomorphism : That Φ is well defined follows by Ka = Kb ⇒ab⁻¹ ∈K = Ker f ⇒f(ab⁻¹) = e' ...