# 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”.