# Statement(s) > [!NOTE] Statement 1 (Group Homomorphism Preserves Inverses) > Let $G,H$ be [[Groups|groups]] and $\phi:G\to H$ a [[Homomorphisms of groups|homomorphism]]. Then for all $g\in G,$ $\phi(g^{-1})=\phi(g)^{-1}.$ # Proof(s) **Proof of statement 1:** Let $g\in G.$ Then it follows from the morphism property and [[Homomorphism of Groups Preserves Identity]] $\phi(g)\phi(g^{-1})=\phi(gg^{-1})=\phi(1_{G})=1_{H}=\phi(1_{G})=\phi(g^{-1}g)=\phi(g^{-1})\phi(g).$Hence by [[Uniqueness of Group Inverses]], $\phi(g^{-1})=\phi(g)^{-1}.$ $\blacksquare$ # Application(s) **Consequences**: **Examples**: # Bibliography