Sylow Subgroup is

Question

G be a finite group.P is a Sylow p-subgroup which is contained in the center of G.Show that there is a normal subgroup N of G such that G=PN.(Herstein problem,page 103 prob 16) Give some idea.P is normal in G,but can't proceed further. Sorry I missed out N intersection P={1}

Answer 1

Check Frattini's Argument, and take into account that

$$Z(G)\lhd G\;\;\;\text{and}\;\;\;N_G(P)=G$$