The question is

Let G be a group, $\displaystyle N \lhd G $ such that $\displaystyle \[G:N\] = r $

let $\displaystyle H \lhd G $ with finite order show that

if

$\displaystyle H \cap N = e $ then $\displaystyle nh = hn \;\; \forall \; h\in H \; , n\in N $