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Group theory

  1. Jun 4, 2007 #1
    1. The problem statement, all variables and given/known data

    Let H and K be subgroups of a finite group G with coprime indices. Prove that G=HK

    2. Relevant equations

    From a theorem we have, If |G:H| and |G:K| are finite and coprime, we have:
    |G:H intersect K|=|G:H|*|G:K|

    |G:H| indicates the index of G over H, not the order here...a notational point that hurts my head.

    3. The attempt at a solution
    I used the thoerem and I got
    |G:H intersect K|=|G:H|*|G:K|

    but since G and H are of coprime index, (H intersect K=1),
    So that I get

    |G|=|G:H|*|G:K|

    if I let |G:H|=p and |G:K|=q then |G|=pq

    That's where I am and I don't think I'm headed in the right direction.

    pointers and clarification will be greatly appreciated.
    CC
     
  2. jcsd
  3. Jun 4, 2007 #2

    NateTG

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    Science Advisor
    Homework Helper

    Well, if you determine that |G:(HK)| =1 you're golden, right?

    BTW:
    Isn't generally true:
    [tex]G = \mathbb{Z}_{12}[/tex]
    [tex]H=\{0,3,6,9\}[/tex]
    [tex]K=\{0,2,4,6,8,10\}[/tex]
    then
    [tex]|G:H|=3[/tex]
    and
    [tex]|G:K|=2[/tex]
    are coprime, but
    [tex]|G \cap K|\neq 1[/tex]
    and
    [tex]|G:H| \times |G:K| \neq |G|[/tex]
     
    Last edited: Jun 4, 2007
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