The image of a group under a homomorphism is a subgroup of the codomain

Let $$f: G \to H$$ be a group ho­mo­mor­phism, and write $$f(G)$$ for the set $$\{ f(g) : g \in G \}$$. Then $$f(G)$$ is a group un­der the op­er­a­tion in­her­ited from $$H$$.

Proof

To prove this, we must ver­ify the group ax­ioms. Let $$f: G \to H$$ be a group ho­mo­mor­phism, and let $$e_G, e_H$$ be the iden­tities of $$G$$ and of $$H$$ re­spec­tively. Write $$f(G)$$ for the image of $$G$$.

Then $$f(G)$$ is closed un­der the op­er­a­tion of $$H$$: since $$f(g) f(h) = f(gh)$$, so the re­sult of $$H$$-mul­ti­ply­ing two el­e­ments of $$f(G)$$ is also in $$f(G)$$.

$$e_H$$ is the iden­tity for $$f(G)$$: it is $$f(e_G)$$, so it does lie in the image, while it acts as the iden­tity be­cause $$f(e_G) f(g) = f(e_G g) = f(g)$$, and like­wise for mul­ti­pli­ca­tion on the right.

In­verses ex­ist, by “the in­verse of the image is the image of the in­verse”.

The op­er­a­tion re­mains as­so­ci­a­tive: this is in­her­ited from $$H$$.

There­fore, $$f(G)$$ is a group, and in­deed is a sub­group of $$H$$.

Parents:

• Group homomorphism

A group ho­mo­mor­phism is a “func­tion be­tween groups” that “re­spects the group struc­ture”.