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Math Help - Finite Group Question

  1. #1
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    Finite Group Question

    Let G be a finite group and let H\neq G be a subgroup of G. Show that G\neq \cup_{a\in G} aHa^{-1}.
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by paulk View Post
    Let G be a finite group and let H\neq G be a subgroup of G. Show that G\neq \cup_{a\in G} aHa^{-1}.
    Hint1: Think about orders.

    Hint 2: Conjugation of a subgroup preserves order.
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  3. #3
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    Let  [G:H]=n\geq 2. Then we have |\cup_{a\in G} aHa^{-1}|\leq n(|H|-1)+1<n|H|=|G|.
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