Prove that the Kernel of a homomorphism is a subgroup of the domain group. Solut

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Let f : G -> H be a homomorphism. If e is the identity of G and e' is the identity of H, then f(e) = e' (It's an easily provable property of a group homomorphism). Thus e is in ker(f), ergo ker(f) is nonempty. Let x,y be in ker(f). Then f(xy) = f(x)f(y) = e'e' = e'. Thus xy is in ker(f) and ker(f) is closed. Let x be in ker(f). Then f(x^(-1)) = f(x)^(-1) = e'^(-1) = e'. Thus x^(-1) is also in ker(f). Therefore ker(f) is a subgroup of the domain.

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