How to determine if a group H is a normal subgroup of group G?

Question

A subgroup $H$ of the group $G$ is normal group in $G$ if and only if $\displaystyle ghg^{-1}\in H$ for all $\displaystyle g\in G$ and $\displaystyle h\in H$. How to use MMA to know the group $H$ is a normal group of group $G$? Such as:

G = SymmetricGroup[6];
H = AlternatingGroup[6];

Answer 1

IsNormalQ[H_, G_] := Sort[DeleteDuplicates[Catenate[
     Outer[PermutationReplace, GroupElements[H], 
      Complement[GroupElements[G], GroupElements[H]]]]], Less] === GroupElements[H]

Or little concise, but sometimes it will be slower:

IsNormalQ[H_, G_] := ContainsExactly[
  Catenate[Outer[PermutationReplace, GroupElements[H], 
    Complement[GroupElements[G], GroupElements[H]]]],GroupElements[H]]

Update(Simplicity and efficiency)

IsNormalQ[H_,G_]:=AllTrue[GroupElements[G],PermutationReplace[H,#]==H&]