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Math Help - Poincare's theorem about subgroups of finite index

  1. #1
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    Poincare's theorem about subgroups of finite index

    Hi:
    I'm given this problem: let G be a group and A, B subgroups of finite order in G. Let D=A\cap B. Prove that u,v \epsilon At\cap Bs \Rightarrow Du=Dv. (1) Using this, prove that ]\leq [G:A][G:B]" alt="[G]\leq [G:A][G:B]" /> (Poincare's theorem).

    I can prove the theorem by defining f: G_D \rightarrow G_A \times G_B, f(Dd)=((Ad, Bd), where G_D is the set of right
    cosets of D in G and so on for G_A and G_B. f is one-to-one and therefor G_D is a finite set. Furthermore, |G_D|\leq |G_A||G_B|.

    My question is: can (1) make the proof simpler or more straightforward than that given by me? (assuming it is correct). Because I've tried to use (1) to prove the theorem but I've failed. Any suggestion would be welcome. Thanks for reading.
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by ENRIQUESTEFANINI View Post
    Hi:
    I'm given this problem: let G be a group and A, B subgroups of finite order in G. Let D=A\cap B. Prove that u,v \epsilon At\cap Bs \Rightarrow Du=Dv. (1) Using this, prove that ]\leq [G:A][G:B]" alt="[G]\leq [G:A][G:B]" /> (Poincare's theorem).

    I can prove the theorem by defining f: G_D \rightarrow G_A \times G_B, f(Dd)=((Ad, Bd), where G_D is the set of right
    cosets of D in G and so on for G_A and G_B. f is one-to-one and therefor G_D is a finite set. Furthermore, |G_D|\leq |G_A||G_B|.

    My question is: can (1) make the proof simpler or more straightforward than that given by me? (assuming it is correct). Because I've tried to use (1) to prove the theorem but I've failed. Any suggestion would be welcome. Thanks for reading.
    Poincare's Theorem states that the intersection of finitely many subgroups of finite index is a subgroup of finite index. So it is sufficient to prove the result for two subgroups, which is what we are doing here.

    Let [tex]D= A \cap B[/MATH and g \in Dh where Dh is some coset of D, g,h \in G.

    \Rightarrow gh^{-1} \in D \Rightarrow g \in Ah \cap Bh \Rightarrow Dh \leq Ah \cap Bh

    Clearly Ah \cap Bh \leq Dh \Rightarrow Ah \cap Bh = Dh.

    Thus, the number of cosets of D is less than or equal to [G:A]*[G:B], and we are done.

    Also, you can put text into your LaTeX by writing "\text{put text here}". For example, g \text{ is an element of } G.
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