Two subgroups $N,H$ if $N\trianglelefteq G$ and $NH\trianglelefteq G$ does that imply $H\trianglelefteq G$? If not give counter example

Question

If $N\trianglelefteq G$ and $H\leq G$, such that $NH \trianglelefteq G$ is it true that $H\trianglelefteq G$?

I guess it is true but am not sure, we know for any $x$ we have $$NH = x^{-1}NHx = x^{-1}Nxx^{-1}Hx = N x^{-1}Hx$$ But that does not mean we can cancel $N$ since it could be the case where $n_1h_1 = n_2 x^{-1}h_2x\Rightarrow n_2^{-1}n_1 h_1 = x^{-1}h_2x$ am not sure how to complete.

Thanks.

Answer 1

The next smallest example is one where $N,H,NH,G$ are all distinct subgroups of $G$.

Consider $G=D_{2\cdot 4}=\langle r,\bar{r}\mid r^4=r\bar{r}r\bar{r}=\bar{r}^2=1\rangle$ the dihedral group of order $8$. Let $N=\{1,r^2\}$, $H=\{1,\bar{r}\}$. Then it is easy to check:

• $N\lhd G$;
• $H<G$ is not normal;
• $NH\lhd G$.

Answer 2

As suggested by @user10354138, we take $S_3=\{1,(12),(13),(23),(123),(132)\}$ then $\langle(123)\rangle \langle (12)\rangle = S_3 \trianglelefteq S_3$ but $\langle (12)\rangle$ not normal.