1

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}

Via
  • 5

1 Answers1

0

Check Frattini's Argument, and take into account that

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

Timbuc
  • 34,191
  • since Z(G) and P both are normal in G we will always get G=G by Frattini argument.It is not clear. – Via Sep 08 '14 at 14:20
  • Frattini's argument says that $;G=N_G(P),Z(G);$ , so... – Timbuc Sep 08 '14 at 14:36
  • But then what is the required N? – Via Sep 08 '14 at 16:52
  • N should be proper normal subgroup of G – Via Sep 08 '14 at 16:53
  • It is written clearly in the link I wrote you, @Via: a normal subgroup $;H;$, which in our case is $;Z(G);$ and a Sylow $;p$-subgroup of $;G;$ contained in $;H;$, which is our $;P;$, then we get $$G=N_G(P)H\stackrel{\text{our case}}=G Z(G)=G$$ since $;P\lhd G\iff H_G(P)=G;$ ...! – Timbuc Sep 08 '14 at 17:38
  • Sorry @ Timbuc I missed it out,I added to the question that P meets N trivially. – Via Sep 08 '14 at 18:17