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Math Help - G cannot be the union of conjugates

  1. #1
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    G cannot be the union of conjugates

    If G is a finite group and H is a subgroup of G then prove that  \displaystyle  G \neq  \bigcup\limits_{a \in G} aHa^{-1}
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  2. #2
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    Quote Originally Posted by Chandru1 View Post
    If G is a finite group and H is a subgroup of G then prove that  \displaystyle  G \neq  \bigcup\limits_{a \in G} aHa^{-1}
    Of course, we assume H is a PROPER subgroup.

    Let G act by conjugation on the set X of all its proper subgroups; then we get that

    s:=|Orb(H)|=[G:N_G(H)]\leq [G:H]=r , say, and since |aHa^{-1}|=|H|\,\,\,\forall\,a\in G ,we get that:

    |\bigcup aHa^{-1}|\leq 1+s(|H|-1)\leq 1+r(|H|-1)=1+r|H|-r=|G|-(r-1)<|G|

    (Question: why |\bigcup aHa^{-1}|\leq 1+s(|H|-1) ??)

    Tonio
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