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Math Help - Question on group theory

  1. #1
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    Question on group theory

    I would like to proof the following in the case of groups of finite order:

    the number of elements of an arbitrary class of conjugated elements is a divider of the index (G:Z) of the centre of G.

    Does anybody know how to proof this?

    Peter Mulder
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  2. #2
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    Quote Originally Posted by PeterMulder View Post
    I would like to proof the following in the case of groups of finite order:

    the number of elements of an arbitrary class of conjugated elements is a divider of the index (G:Z) of the centre of G.

    Does anybody know how to proof this?

    Peter Mulder

    Let a group G act on itself by conjugation: G\times G\rightarrow G\,,\,g\cdot x\to gxg^{-1} , then |Orb(x)|=[G:Stab(x)] ,

    with Stab(x):=\{g\in G/\,gxg^{-1}=x\}=C_G(x)= the centralizer of x in G

    Since Orb(x) is the equivalence class of x and since Z(G)\leq C_G(x) , we're done by Lagrange's theorem

    Tonio
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  3. #3
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    Thank you Tonio!
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