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' ⇒f(a) (f(b))⁻¹ = e' ⇒f(a) = f(b) ⇒Φ(Ka) = Φ(Kb) By retracing the steps backward , we will prove that Φ is one _one . Again as Φ(KaKb) = Φ(Kab) = f(ab) = f(a)f(b) = Φ(Ka) Φ(Kb) we find Φ is a homomorphism . To Check That Φ Is Onto : Let g' ∈ G' be any element . Since