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The image of a group homomorphism is a subgroup of the codomain
Proposition (Image is a subgroup). Let be a group homomorphism. Let be its image. Then is a subgroup of .
Context. Together with "kernel is normal," this gives the basic structural picture of any homomorphism: it factors through a quotient of onto a subgroup of .
Let be groups. A group homomorphism is a function such that for all ,
(Here denotes the product in and the product in .)
Basic consequences of the defining identity include: and for all . Two fundamental associated subgroups are the kernel and the image of . These are tied together by the first isomorphism theorem.