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Homework Help: Centralizers and elements of odd order

  1. Mar 22, 2010 #1
    I was going over a proof in which the following is given:

    if abb=bba and b is of odd order, then ab=ba (i.e. if b^2 centralizes a then so does b)

    I'm not sure why this is so. Any clarification would be appreciated.

    Cheers,
    W. =)
     
  2. jcsd
  3. Mar 22, 2010 #2

    Dick

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    b^2 commutes with a. If b has odd order then b^(2n+1)=e for some n. So b^(2n+1)a=a(b^(2n+1)). Do you see it now?
     
  4. Mar 22, 2010 #3
    I believe so...
    b^(2n)ba = ab^(2n)b
    b^2n(ba) = b^2n(ab) since abb = bba
    ba = ab

    If that's it, then cheers! =)
     
  5. Mar 22, 2010 #4

    Dick

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    Cheers!
     
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