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Suppose G and H are groups and ϕ:G -> H is a homomorphism. Let N be...

Suppose G and H are groups and ϕ:G -> H is a homomorphism. Let N be a normal subgroup of G contained in ker(ϕ). Define a mapping ψ: G/N -> H by ψ (aN)= ϕ (a) for all a in G.

Prove that ψ is a well-defined homomorphism from G/N to H.

Is ψ always an isomorphism? Prove it or give a counterexample

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