Abstract Algebra/Group Theory/Subgroup/Normal Subgroup/Definition of a Normal Subgroup

A normal subgroup is a subgroup H of a group G that satisfies

${\displaystyle \forall \;g\in G:gHg^{-1}=H}$
where ${\displaystyle gHg^{-1}=\lbrace g\ast h\ast g^{-1}|h\in H\rbrace }$

Equivalent Definition

1. ${\displaystyle \forall \;g\in G,h\in H:g\ast h\ast g^{-1}\in H}$

Proof

 ${\displaystyle gHg^{-1}\subseteq H}$ by this definition
${\displaystyle H\subseteq gHg^{-1}}$
 0. Choose ${\displaystyle g\in G,x\in H}$ 1. ${\displaystyle g^{-1}\in G}$ 2. ${\displaystyle g^{-1}\ast x\ast [g^{-1}]^{-1}\in H}$ by this definition 3. ${\displaystyle g\ast (g^{-1}\ast x\ast [g^{-1}]^{-1})\ast g^{-1}\in gHg^{-1}}$ 4. ${\displaystyle h\in gHg^{-1}}$