# Math Help - normal subgroups

1. ## normal subgroups

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

Is $N_k$ normal to $G$, i.e., $N_k \lhd G$ ? 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 $N_k$ normal to $G$, i.e., $N_k \lhd G$ ? why?
A counterexample is,

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

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

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