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.(adsbygoogle = window.adsbygoogle || []).push({});

1) If p divides |H| then P[itex]\cap[/itex]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[itex]\cap[/itex]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.

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# Product of two subgroups and intersection with p-subgroup

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