Proof of Sylow: Let G be a Finite Group, H and K Subgroups of G

[moont14263]

Let G be a finite group, H and K subgroups of G such that G=HK. Show that there exists a p-Sylow subgroup P of G such that P=(P∩H)(P∩K).

I found this proof and it is clear http://math.stackexchange.com/questions/42495/sylow-subgroups but I do not understand step 3 which is "It is clear in this situation that P=(P∩H)(P∩K)". Help.

 

[DonAntonio]

Let G be a finite group, H and K subgroups of G such that G=HK. Show that there exists a p-Sylow subgroup P of G such that P=(P∩H)(P∩K).

I found this proof and it is clear http://math.stackexchange.com/questions/42495/sylow-subgroups but I do not understand step 3 which is "It is clear in this situation that P=(P∩H)(P∩K)". Help.

+

Well, after part (3) in the stackexchange there's a further poster who also had a problem with this step, and I think it isn't THAT clear, as the other guy wrote, that P = (P/\H)(P/\K)...
The same question as that further poster asked came to my mind: is normality something we can do without?

It'd be interesting to know where did you find this problem.

DonAntonio

 

[moont14263]

It is in Finite Group Theory I. Martin Isaacs page 18