If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$. Exercise 11, page 45 from Hungerford's book Algebra.

If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every
  subgroup of $H$ is normal in $G$.

I am trying to show that $a^{-1}Ka\subset K$, but I got stuck. What I am supposed to do now?
Thanks for your kindly help.

since $H$ is normal in $G$ you get $a^{-1}Ka \subset H$, for all $a\in G$. Now use the fact that $H$ is cyclic (there is only one subgroup of $H$ such that $\dots$)
