Let P be a p-sylow sbgrp of a finite group G.

N(P) will be the normalizer of P in G. The quotient group N(P)/P is cyclic from order n.

PROVE that there is an element a in N(P) from order n and that every element such as a represnts a generator of the quotient group N(P)/P

My try:

Welll.... there is mP in N(P)/P such as (mP)^n = P -> m^n*P=P -> m^n is in P...P is a p-sylow sbgrp so m^n must have order p^r where r is a natural number. Hence the element m^(p^r) which is obviously in N(P) has order n... But why it represents a generator of N(P)/P ?

TNX in advance