# Normal subgroup of a normal subgroup of a group is not always a normal subgroup of a group. Prove using GAP. [duplicate]

How could I show that a normal subgroup of a normal subgroup of a group is not always a normal subgroup of a group?

Here is what I did. I know the Algebra system GAP, so I wrote the following function:

performTheProve:= function()
local g, gDash, AFour, NormalOfAFour, NormalK;
AFour:=AlternatingGroup(4);
NormalOfAFour:=NormalSubgroups(AFour);
for g in  NormalOfAFour do
for gDash in NormalSubgroups(g) do
if(not gDash in NormalOfAFour) then
return [gDash, g];
fi;
od;
od;
end;


This function produces the following result:

[ Group([ (1,2)(3,4) ]), Group([ (1,3)(2,4), (1,2)(3,4) ]) ]


So, now I am stating that the Group([ (1,3)(2,4), (1,2)(3,4) ]) is a normal subgroup of a ${A}_{4}$, Group([ (1,2)(3,4) ]) is a normal subgroup of a Group([ (1,3)(2,4), (1,2)(3,4) ]) and not a normal subgroup of a ${A}_{4}$.

Given this example I am proving that normal subgroup of a normal subgroup of a group is not always a normal subgroup of a group. Am I right?