Prove the First Isomorphism Theorem for Groups
Let φ:G → G̅ be a group homomorphism. We want to prove the First Isomorphism Theorem in Group Theory, which says that the factor group (quotient group) G/Ker(φ) is isomorphic to the direct image φ(G). We do this by defining a mapping ψ:G/Ker(φ) → φ(G) with the equation ψ(g Ker(φ)) = φ(g). We must prove: 1) ψ is well-defined, 2) ψ is one-to-one, 3) ψ is onto, and 4) ψ is operation-preserving.
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