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Math Help - Conjugacy and Abelian Groups

  1. #1
    Super Member Bernhard's Avatar
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    Conjugacy and Abelian Groups

    Beachy and Blair in the book Abstract Algebra section 7.2 Conjugacy define conjugacy as follows:

    Let G be a group and let x, y \in G. The element x is said to be a conjugate of the element x if there exists an a \in G such that y = {ax}a^{-1}

    They then make the statement that "in an abelian group, elements or subgroups are only conjugate to themselves"

    I have seen some brief justifications of this statement in other texts but cannot see how they link to the definition.

    Can someone please - starting from the definition and proceeding from there - give a formal proof of this statement and then go on to show that (asserted in many texts) the conjugacy classes of an abelian group are all singleton sets>

    Peter
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  2. #2
    MHF Contributor Amer's Avatar
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    Re: Conjugacy and Abelian Groups

    let G be abelian to find the conjugacy class of any element x we find

    g x g^{-1} = gg^{-1} x = x \;\; for any g in G

    or to the definition suppose that y is conjugate to x that means there exist g in G such that

     y = g x g^{-1} but G is abelian  y = gg^{-1} x \Rightarrow y =x

    if an element x is conjugate to itself that mean x in the center of G
    and since G is abelian the center of G is G itself Z(G) = \{ g \in G \mid g.a = a \;\;\;\forall a \in G \}
    in our case Z(G) = \{ g \in G \mid gag^{-1} = a \;\; \forall a\in G \}
    and since G abelian all g elements are in Z(G) (i.e all elements are conjugate to itself )
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