Is $N$ normal to $HN$ if $H$ subgroup and $N$ normal subgroup of the group $G$ If $H$ is a subgroup of $G$ and $N$ is a normal subgroup of $G$, then whats the relation between $N$ and the subgroup $HN$ in respect to normality, i.e. must $N$ be normal to $HN$ ?

Given:


*

*$G$ is a group.

*$H\lt G$.

*$N$ normal to $G$.


To Show: $N$ normal to $H\circ N$.
Possible Proof:
Now,
$H\circ N$ = {$h\circ n | h\in H, n\in N$}$ = H\cup N$.
Since, $H\lt G$ and $N$ normal to $G$ $\implies H, N \lt G \implies (H\cup N)\subseteq G\implies H\circ N\subseteq G$. 
Now,
$N$ normal to $G$ 
$\iff g\circ N\circ g$-1 $= N, \forall g\in G$. [This comes from the basic properties of a normal subgroup.]
$\implies g\circ N\circ g$-1 $= N, \forall g\in H\circ N$. [Since, $H\circ N\subseteq G$, already proved.]
$\iff N$ normal to $H\circ N$.

QED

PS: Please let me know if there are any errors in the proof. I myself wanted a proof for this theorem.
