If $G$ is a (non-abelian) $p$-group, $|G|=p^n$, $n>3$, then it is elementary that $G$ contains a (normal) abelian subgroup of order $p^2$. It is also true that $G$ necessarily contains a **normal abelian subgroup** of order $p^3$ (*Group Theory - W. R. Scott*).

**1)** What is the largest possible value of $m$ such that **any** non-abelian group of order $p^n$ contains a **normal abelian subgroup** of order $p^m$?

**2)** What is the largest possible value of $m$ such that **any** non-abelian group of order $p^n$ contains an **abelian subgroup** of order $p^m$?

[Please suggest references.]