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

A normal subgroup is a [[../../Definition of a Subgroup|subgroup]] H of a group G that satisfies

$\mathrm{\forall }g\in G:gH{g}^{-1}=H$

where $gH{g}^{-1}=\{g\ast h\ast g^{-1}|h\in H\}$

Equivalent Definition

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

Proof

$gH{g}^{-1}\subseteq H$

by this definition

$H\subseteq gH{g}^{-1}$

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. $g\ast (g^{-1}\ast x\ast [g^{-1}{]}^{-1})\ast g^{-1}\in gH{g}^{-1}$
4. $h\in gH{g}^{-1}$