RO-NMO 2014 Problem 124

Let be a finite group $G$ that has an element $a\neq 1$ for which exists a prime number $p$ such that $x^{1+p}=a^{-1}xa,$ for all $x\in G.$ a) Prove that the order of $G$ is a power of $p.$ b) Show that $H:=\{x\in G|\text{ord} (x)=p\}\le G$ and $\text{ord}^2(H)>\text{ord}(G).$