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Index of subgroups

  1. Sep 30, 2008 #1
    1. The problem statement, all variables and given/known data
    Let G be a group and let H,K be subgroups of G.
    Assume that G is finite and that the indices |G:H| and |G:K| are relatively prime. Show that G=HK.

    Hint: Show that |G:H(intersect)K| is divisible by both |G:H| and |G:K| and then use the counting principle for |HK|.


    3. The attempt at a solution

    First off, why do the indices have to be relatively prime?
    I don't know how to show that |G:H(intersect)K| is divisible by both |G:H| and |G:K|, but I do know that if I assume those, I know how to use the counting principle because ultimately it will come down to saying that |HK|=c|G| for some multiple c, and c must = 1 otherwise it says that for c>1, |HK|>|G| and that is not possible.

    EDIT:
    Is the intersection of the left coset of H and the left coset of K disjoint? Since they are both equivalence classes they would have to either be disjoint or equal, no? So then |G:H(intersect)K| would consist of both xH and xK for some x in G.....?
     
    Last edited: Oct 1, 2008
  2. jcsd
  3. Oct 1, 2008 #2

    morphism

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    If the indices were not prime then it's easy to come up with examples where G does not equal HK.

    As to showing that |G:H[itex]\cap[/itex]K| is divisible by both |G:H| and |G:K|, here's a hint: H[itex]\cap[/itex]K is a subgroup of H, K and G.
     
  4. Oct 1, 2008 #3
    I understand that H(union)K is a subgroup of H, K and G. But I don't understand how the numbers would work. How do we know that |G:H(union)K| is definitely a multiple of both |G:H| and |G:K|?
     
  5. Oct 1, 2008 #4
    I just added this "edit" into my original question:

    Is the intersection of the left coset of H and the left coset of K disjoint? Since they are both equivalence classes they would have to either be disjoint or equal, no? So then |G:H(intersect)K| would consist of both xH and xK for some x in G.....?
     
  6. Oct 1, 2008 #5

    morphism

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    Just write down what |G:H|, |G:K| and |G:H[itex]\cap[/itex]K| are. It will also help to think about what |H:H[itex]\cap[/itex]K| and |K:H[itex]\cap[/itex]K| are.

    "The" left coset of H? I think you need to review your definitions. A coset is not an equivalence class; it's a set.
     
    Last edited: Oct 1, 2008
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