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Projection homomorphism: normal subgroup

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Let M be a normal subgroup of a group G and let N be a normal subgroup of a group H. Use the first Isomorphism Theorem to prove that M X N is normal subgroup of G X H and that (G X H) / (M X N) is isomorphic to G/M X H/N

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This solution clearly assesses the projection homomorphism.

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Since M is a normal subgroup of G, there's the projection homomorphism p: G -> G/M given by p(g) = gM, and M=ker(p).
Since N is normal in H, there's the projective homomorphism q: H -> H/N given by q(h) = hN, and N = ker(q).

Define a map s: G X H -> G/M ...

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