Fundamental Theorem Of Homomorphism Of Group
Fundamental Theorem Of Homomorphism Of Group : Let Φ be a homomorphism of G onto G̅ with kernel K .Then G/K ≈ G̅ . Proof: Let G̅ be the homomorphie image of a group G and Φ be the corresponding homomorphie . Then K is normal subgroup of G . To prove that G/K ≈ G̅ . If a ∈ G , the Ka ∈ G /K and Φ(a) ∈G̅ Let ψ : G/K →G̅ such that ψ(Ka) = Φ(a) ∀ a∈G Where Ka is called right coset and Kb is called left coset. To Show The Mapping ψ is well defined : i.e if a,b ∈G and Ka = Kb ,then ψ(Ka) = ψ(Kb) We have Ka = Kb => ab⁻¹ ∈ K => Φ(ab⁻¹) = e̅ (identity of G̅) =>Φ(a) Φ(b⁻¹) = e̅ => Φ(a) [Φ(b)]⁻¹ = e̅ => Φ(a) [Φ(b)]⁻¹Φ(b) = e̅ Φ(b) => Φ(a) e̅ = Φ(b) => Φ(a) = Φ(b) => ψ(Ka) = ψ(Kb) => ψ is well defined To Prove ψ is one _one : We have ψ(Ka) =