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Product of two subgroups and intersection with p-subgroup

  1. Mar 14, 2012 #1
    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[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.
  2. jcsd
  3. Mar 14, 2012 #2
    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.
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