1. ## normal subgroups

I have $\displaystyle N_k$ is a normal subgroup of $\displaystyle K$ while $\displaystyle K$ is a normal subgroup of $\displaystyle G$.
i.e., $\displaystyle N_k \lhd K \lhd G$.

Is $\displaystyle N_k$ normal to $\displaystyle G$, i.e., $\displaystyle N_k \lhd G$ ? why?

2. i found a counter example after posting my reply..
but,.. after i got home, alice had posted the counter example i was thinking...

3. Originally Posted by deniselim17
Is $\displaystyle N_k$ normal to $\displaystyle G$, i.e., $\displaystyle N_k \lhd G$ ? why?
A counterexample is,

Consider $\displaystyle D_4=\{1, r, r^2, r^3, s, rs, r^2s, r^3s\}$.

$\displaystyle D_4$ has a subgroup isomorphic to a Klein-4 group, which is $\displaystyle \{1, r^2, s, r^2s\}$.
A normal subgroup of $\displaystyle \{1, r^2, s, r^2s\}$ is $\displaystyle \{1,s\}$. However, $\displaystyle N=\{1,s\}$ is not normal in $\displaystyle D_4$, since $\displaystyle rN \neq Nr$.

4. Here is another counterexample.
Consider $\displaystyle S_4$ and $\displaystyle V = \{ \text{id},(12)(34),(13)(24),(14)(23)\}$ and let $\displaystyle H = \{ \text{id},(12)(34)\}$.
We see that $\displaystyle V\triangleleft S_4\text{ and }H\triangleleft V$ but $\displaystyle H\not \triangleleft S_4$.