# Image of the identity under a group homomorphism is the identity

For any group ho­mo­mor­phism $$f: G \to H$$, we have $$f(e_G) = e_H$$ where $$e_G$$ is the iden­tity of $$G$$ and $$e_H$$ the iden­tity of $$H$$.

In­deed, $$f(e_G) f(e_G) = f(e_G e_G) = f(e_G)$$, so pre­mul­ti­ply­ing by $$f(e_G)^{-1}$$ we ob­tain $$f(e_G) = e_H$$.

Parents:

• Group homomorphism

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