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### Group Homomorphism

A group homomorphism is a map between two groups such that the group operation is preserved: for all , where the product on the left-hand side is in and on the right-hand side in .As a result, a group homomorphism maps the identity element in to the identity element in : .Note that a homomorphism must preserve the inverse map because , so .In particular, the image of is a subgroup of and the group kernel, i.e., is a subgroup of . The kernel is actually a normal subgroup, as is the preimage of any normal subgroup of . Hence, any (nontrivial) homomorphism from a simple group must be injective.

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