TITLE: Using the properites of normal subgroups to show that two elements commute
QUESTION [5 upvotes]: $Question - $ Let $N, H$ be normal subgroups of a group $G$ such that $N\cap H = \{e_G\}$. Show that, for all $g\in N$ and $h\in H$, $gh=hg$.
If $N, H$ are normal subgroups, I know that $aN = Na, aH = Ha$ for each $a\in G$. I tried to write $gh$ = $g.e_G.h$, but I don't think this can help me. I also know that for each $a\in G$ we have $a^{-1}Na = N, a^{-1}Ha = H$, so I wrote $gh = g.a.a^{-1}.g = g.a.e_G.a^{-1}.g$, but I think this can't help me as well.
I'd appreciate if someone could help me!

REPLY [9 votes]: $$ghg^{-1}h^{-1}= (ghg^{-1})h^{-1}\in H$$
$$ghg^{-1}h^{-1}= g(hg^{-1}h^{-1})\in N$$
Therefore $ ghg^{-1}h^{-1}\in N\cap H$, hence $ ghg^{-1}h^{-1}=e$.