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Homework Help: Abstract Algebra Proof (Cyclic cycles & order)

  1. Nov 22, 2013 #1
    Prove that if G is a group and aεG, then o(a-1)=o(a)

    This is all I have so far:
    Assume G is a group and aεG. Because G is a group a has an inverse in the group, a-1 s.t. aa-1=e, which is also in G.
    <a>={an|nεZ}. |<a>| is the number of elements in <a> before it cycles back.

    Basically all I've done is write what I know about what is given. I have tried to find <a> and <a-1> from a previous problem to see if there is a pattern but I don't see one.
  2. jcsd
  3. Nov 22, 2013 #2
    Can you prove ##<a> = <a^{-1}>##?
  4. Nov 22, 2013 #3


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    Or can you show ##(a^n)^{-1}=(a^{-1})^n## and use that to show ##a^n=e## if and only if ##(a^{-1})^n=e##?
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