A group homomorphism induces a subgroup homomorphism

Given a Group homomorphism and a subgroup , the map is also a homomorphism. #m/thm/group Quite trivial.

Proof

is a group homomorphism iff for all . Clearly if , the property still holds, hence is a homomorphism.

Corollary

It follows that a group automorphism establishes isomorphisms between subgroups, since the the image of a group homomorphism is a subgroup, which for an automorphism will be a subgroup of equal size.

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