Let G be a group. If H is a normal subgroup in G, and K is a normal subgroup of H, when is K a normal subgroup of G?

Question

I know that K is normal in G iff K is characteristic in H. However how can you prove this, and how do you show that K is normal in H is not sufficient for K to be normal in G?

Answer 1

Let $G=A_4, H=\{(1\;2)(3\;4),(1\;3)(2\;4),(1\;4)(2\;3),e\}, K=\{e, (1\;2)(3\;4)\}$. Then $H$ is normal in $G$ and $K$ is normal in $H$. But $K$ is not normal in $G$.

Answer 2

There are some cases where normality is transitive. Let $\;K\lhd H\lhd G\;$ , then $\;K\lhd G\; $ if

$$\begin{align*}(1)&\;\;K\,\text{char.}\,\\ (2)&\;\; H\;\;\text{is cyclic}\\ (3)&\;\;H\;\;\text{is a direct factor of}\;\;G\end{align*}$$

Groups in which normality is transitive are called $\;T$-groups, and there's a reasonably wide literature on this.