Suppose $H,N\leq G$, $N\unlhd G$, $H$ and $N$ have trivial intersection, and $HN=G$.
I want to show that if $G\approx H\times N$ (isomorphic), then $H\unlhd G$. What I do is identify $H$ with $H'=\{(h,1)|\ h\in H\}$, and try to show $H'\unlhd H\times N$. I get $(h',n)(h,1)(h'^{-1},n^{-1})=(h'hh'^{-1},1)\in H'$, so it seems like $H'\unlhd H\times N$. I just feel this is taking a leap in logic. Is there a more formal/more correct way to get this implication?