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

$\forall \;g\in G:gHg^{-1}=H$

where $gHg^{-1}=\lbrace g\ast h\ast g^{-1}|h\in H\rbrace$

Equivalent Definition

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

Proof

$gHg^{-1}\subseteq H$

by this definition

$H\subseteq gHg^{-1}$

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