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

 * {| Style = "width:70%"


 * $$ gHg^{-1} \subseteq H $$
 * by this definition
 * }
 * }


 * $$ H \subseteq gHg^{-1} $$
 * {|Style = "width:70%"

|-    |-
 * 0. Choose $$ g \in G, x \in H $$
 * 1. $$ g^{-1} \in G $$
 * 1. $$ g^{-1} \in G $$

|-
 * 2. $$ g^{-1} \ast x \ast [g^{-1}]^{-1} \in H $$
 * 2. $$ g^{-1} \ast x \ast [g^{-1}]^{-1} \in H $$

|-
 * by this definition
 * 3. $$ g \ast (g^{-1} \ast x \ast [g^{-1}]^{-1}) \ast g^{-1} \in gHg^{-1} $$
 * 3. $$ g \ast (g^{-1} \ast x \ast [g^{-1}]^{-1}) \ast g^{-1} \in gHg^{-1} $$

|-
 * 4. $$ h \in gHg^{-1} $$
 * 4. $$ h \in gHg^{-1} $$


 * }
 * }