There is no core free subgroup of order $p^2$ in a group of order $p^4$ By the classification of group of order $p^4$ ($p$ odd prime) from Burnside's book it seems to me that there is no core free subgroup of order $p^2$ in a group of order $p^4$. If I am not wrong there must be some argument to prove this. Can someone help me in proving this? 

This is a partial answer.
Let $K<H<G, |K|=p^2, |H|=p^3$. Suppose that $H$ is non-abelian. Since $K$ is normal in $H$, then $Z(H)<K$ (otherwise $H=Z(H)\times K$, what is impossible). Further, $Z(H)$ is characteristic in $H$, hence it is normal in $G$. So the core of $K$ is not trivial.
