Normality of a normal subgroup of normal subgroup of G

Question

Let $G$ be a non-Abelian Group and $H$ is normal subgroup of $G$. Is it always true that a normal subgroup $K$ of $H$ is also normal in $G$? Justify your answer.

My answer is that, this is not true in general. I created a counter example. Consider on $A_4$,{ Group of even permutation of four symbols}, $K_4$ {Klein's four group} is normal subgroup of $A_4$, Now take any subgroup of $K_4$, subgroup will be normal in $K_4$ {being Abelian group }, But this subgroup will not be normal in $A_4$,as we know that proper normal subgroup of $A_4$ is only $K_4$.

Now my question:

1) am I correct?

2) Can we prove it instead of giving counter example?

3) Different counter examples are also invited.

Thank you.

Answer 1

1) Yes you're correct

2) Giving a counterexample is a "proof"

3) to give a more natural reason, we have $K\triangleleft H$ means $hK=Kh$ for all $h\in H$. There is no reason to assume that this should extend to all elements of $G$; that is, you should not assume that $gK=Kg$ for all $g\in G$ just because it is true for all $g\in H$.

Also to add a bit, if I'm not mistaken, the group $K$ you are speaking of has a property called being "semi-normal" in $G$. My advisor told me a bit about this but I honestly haven't done any looking into it of my own; but maybe you could look up some stuff on this.

Answer 2

1)Yes, you are correct Chiranjeev.

2)No, I don't think it can be proved in general. For example in your counterexample $V_4$ is a normal subgroup of itself, and so it is clearly a normal subgroup of $V_4$. If you refined your statement by stipulating some extra conditions then maybe.