The image of a group homomorphism is a subgroup

Let and be groups, and be a Group homomorphism. Then the image

is a subgroup of . #m/thm/group

Proof

Since , clearly . Let . Then there exist (not necessarily unique) such that and . It follows that . Therefore is a subgroup by One step subgroup test.

Corollary

It follows that the image of a subgroup is also a subgroup, since a group homomorphism induces a subgroup homomorphism.

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