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Groups : Isomorphism and Homomorphism

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Show that a group G is simple if and only if every nontrivial group homomorphism G -> G1 is one-to-one.

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If G is simple, then G has no nontrivial normal subgroup. Then for any nontrivial group homomorphism f: G->G1, by the isomorphism theory, we have ...

Solution Summary

Group simplicity is investigated.

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Homomorphisms are examined.

EDIT: G(p) = {x in G : |x| = p^k for some k greater than or equal to 0}

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