Show $H$ is normal under some conditions

Question

Problem
Let $k,m$ be positive integers and $p$ be a prime number such that $k$ and $p^2-p$ are relatively prime. A group $G$ of order $kp^m$ has subgroups $N,H$ satisfying the following conditions.
(i)$N$ is a cyclic group of order $p^m$ and normal in $G$.
(ii)H is of order $k$.
Show that $G=N\times H$.

I tried to show this problem. I found that it suffices to show $H$ is normal in $G$ because $G=NH$, $N$ is normal, and $N\cap H=\{e\}$ ($\because k,p$ are relatively prime).

What shoud I do to prove $H$ is normal?

Answer 1

Hint: $G/C_G(N)$ injects homomorphically in $Aut(N)$. Show that the image of $H$ is trivial, whence $H$ centralizes $N$. Since $G=HN$ it follows that $H$ is normal.