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Math Help - equivalence relation in a group

  1. #1
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    equivalence relation in a group

    Let G be a group. Prove that the relation R on G, defined by xRy if and only if there exists an a that belongs to G such that a^-1*x*a, is an equivalence relation.
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    Quote Originally Posted by anitra View Post
    Let G be a group. Prove that the relation R on G, defined by xRy if and only if there exists an a that belongs to G such that a^-1*x*a, is an equivalence relation.
    you mean xRy \Leftrightarrow y=a^{-1}xa?

    Do you know what an equivalence relation is? Which of the three properties are you having trouble proving?
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    Yes. that's exactly what I mean.
    I do know what an equivalence relation is, but I got stuck at proving the reflexivity.


     x=a^{-1}xa
    is this trivial? Or how would I prove that?
    Once I understand how to prove that, I think I'd be able to prove symmetry and transitivity.
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    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by anitra View Post
    Yes. that's exactly what I mean.
    I do know what an equivalence relation is, but I got stuck at proving the reflexivity.


     x=a^{-1}xa
    is this trivial? Or how would I prove that?
    Once I understand how to prove that, I think I'd be able to prove symmetry and transitivity.
    Can you think of an element a \in G for which  x=a^{-1}xa ? Remember, you just have to show that there exists such an a, not that any choice of a works (which, generally, is false).
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    ah, you got me there, you see, I'm having massive difficulty with group theory..
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  6. #6
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    Quote Originally Posted by anitra View Post
    Yes. that's exactly what I mean.
    I do know what an equivalence relation is, but I got stuck at proving the reflexivity.


     x=a^{-1}xa
    is this trivial? Or how would I prove that?
    Once I understand how to prove that, I think I'd be able to prove symmetry and transitivity.
    You don't prove it. It isn't always true. What you want to prove is that there exist x such that x= a^{-1}xa.

    What if a= e?
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