# Statement(s) > [!NOTE] Statement 1 ($H \unlhd G$ iff $\forall g\in G: gHg^{-1}\subset H$ ) > Let $H$ be a [[Subgroup|subgroup]] of a [[Groups|group]] $G.$ Then $H$ is [[Normal Subgroup|normal]] iff for all $g\in G,$ $gHg^{-1} \subset H.$ # Proof(s) **Proof of statement 1:** Suppose for all $g\in G,$ $gHg^{-1}\subset H.$ Since $g^{-1} \in G,$ we have $g^{-1}Hg \subset H.$ Left-multiplying by $g$ and right-multiplying by $g^{-1}$ gives $H=g(g^{-1}Hg)g^{-1} \subset gHg^{-1}.$ Hence $gHg^{-1}=H.$ $\blacksquare$ # Application(s) **Consequences**: **Examples**: # Bibliography