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Let G be a group and α : G → H be a homomorphism of groups...

Let G be a group and α : G → H be a homomorphism of groups with H abelian. Show that α factors via G/[G, G], i.e. there exists a homomorphism β : G/[G, G] −→ H, such that α = β◦q, where q : G −→ G/[G, G] is the quotient homomorphis

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