# 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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#### Solution Preview

Proof:

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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