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Math Help - Simple group

  1. #1
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    Simple group

    Hi,

    Problem: G - simple \Rightarrow has 8 subgroups of order 7 and we could embedding it into S_8.

    Solution: First part is easy to show from Sylow Theorem.
    Suppose that we have H_1,...,H_8 - Sylow 7-subgroups and consider following mapping:
    \phi : G \to S(\{H_1,...,H_8\})\cong S_8
    \phi(g)(H_i)=gH_ig^{-1}, g \in G and i=1,...,8.
    \phi is homomorphism (it is easy to show that \phi(gh)(H_i)=(\phi(g)\circ\phi(h))(H_i))

    \mbox{Ker}\phi=\{g\in G : \forall_{i=1,...,8} \phi(g)(H_i)=H_i\}=\{g\in G : \forall_{i=1,...,8} gH_ig^{-1}=H_i\} <--- may I describe kernel such that? If yes then:

    \mbox{Ker}\phi=\{1\} or \mbox{Ker}\phi=G - because G is simple.

    If \mbox{Ker}\phi=G, then gH_i=H_ig, so H_1,...,H_8 are simple - bad choice.

    So then \mbox{Ker}\phi=\{1\} \rightarrow \phi is monomorphis. [qed]

    Is this solution good? Thanks for any advices.
    Last edited by Arczi1984; December 5th 2010 at 09:16 PM.
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  2. #2
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    Quote Originally Posted by Arczi1984 View Post
    Hi,

    Problem: G - simple \rightarrow has 8 subgroups of order 7 and we could embedding it into S_8.

    Solution: First part is easy to show from Sylow Theorem.
    Suppose that we have H_1,...,H_8 - Sylow 7-subgroups and consider following mapping:
    \phi : G \to S(\{H_1,...,H_8\})\cong S_8
    \phi(g)(H_i)=gH_ig^{-1}, g \in G and i=1,...,8.
    \phi is homomorphism (it is easy to show that \phi(gh)(H_i)=(\phi(g)\circ\phi(h))(H_i))

    \mbox{Ker}\phi=\{g\in G : \forall_{i=1,...,8} \phi(g)(H_i)=H_i\}=\{g\in G : \forall_{i=1,...,8} gH_ig^{-1}=H_i\} <--- may I describe kernel such that? If yes then:

    \mbox{Ker}=\{1\} or \mbox{Ker}=G - because G is simple.

    If \mbox{Ker}=G, then gH_i=H_ig, so H_1,...,H_8 are simple - bad choice.

    So then \mbox{Ker}=\{1\} \rightarrow \phi is monomorphis. [qed]

    Is this solution good? Thanks for any advices.
    A ""tiny"" omission in your post : you don't say what group G is!!

    Tonio
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  3. #3
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    You're right

    Let G be a group of order 168.

    Like You wrote it was "tiny" omission My eye, You punched me
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  4. #4
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    Quote Originally Posted by tonio View Post
    A ""tiny"" omission in your post : you don't say what group G is!!

    Tonio


    As \ker\phi=N_G(H)\,,\,\,N_G(H)=\ker\phi=G\Longleftri  ghtarrow H\triangleleft G , which is absurd as we're assuming, I hope,

    that G is simple...

    Tonio
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