Finite group with the property that every element of order a power of $p$ is contained in a conjugacy class of size a power of $p$ 
Let $p$ be a prime and let $G$ be a finite group with the property (*)
   that every element of order a power of $p$ is contained in a conjugacy
  class of size a power of $p$.
i) If $p\mid\left|G\right|$ prove that $G$ has a central subgroup $Z$
   of order $p$.
ii) If $Z$ is as above, prove that $G/Z$ has property (*).
iii) Deduce that a Sylow $p$-subgroup of $G$ is normal.

I don't have idea to start. Help me.
Thanks a lot.

Let $P \in {\rm Syl}_p(G)$. Since nontrivial $p$-groups have nontrivial centres, we can choose $g \in Z(P)$ of order $p$. Then $P \le C(g)$, so the conjugacy class of $g$ has order dividing $|G|/|P|$, which is coprime to $p$, so it must be $1$ by hypothesis, and hence $g \in Z(G)$. In fact $Z(P) \le Z(G)$.
