This shouldn't be too hard but I'm having a hard time putting a few pieces together. Let [itex]G[/itex] be a finite group with [itex]N[/itex] a normal subgroup. Let [itex]P[/itex] be a Sylow [itex]p[/itex]-subgroup of [itex]N[/itex]. Prove that [itex]G=N_G(P)N[/itex], where [itex]N_G(P)[/itex] denotes the normalizer of [itex]P[/itex] in [itex]G[/itex]. My attempts: I know that the number of conjugates of P in G equals the index of the normalizer of P in G. What I don't understand is: why does the number of conjugates of P in G equal the normal subgroup N?