Product of two subgroups and intersection with p-subgroup

[moont14263]

Let G be a finite group. P is a Sylow p-subgroup of G and K is normal in G also H is a subgroup of G with (|K|,|H|)=1.
1) If p divides |H| then P$\cap$HK is a subgroup of H.
2) Is (1) when K is not normal in G.

This is my try of (1);
Let y be an element of P$\cap$HK, --> |y| divides |HK|=|H|*|K|--> |y| divides |H|--> y is an element of H as the order of y does not divide the order of K. What I am saying is that the elements of p power order of the intersection come from the Sylow p-subgroups of H.

For (2);
I know that |HK|=|H|*|K| but HK may not be a subgroup of G. And in this case also the intersection may not be a subgroup of G.

Help,me please.

 

[moont14263]

I can see that S_3 is a counter example for (1) and so there is no need to check (2). Thank you very much.