Let G be a group of order $ap^{n}$ where p is prime and (a,p)=(a,p-1)=1.Suppose that some Sylow p-subgroup $\, P
we can consider the homomorphism $N(P)\rightarrow Aut(P)$,then the kernel is the centralizer of P, but I haave no idea of what is going on, can anyone help me with this?