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Disjoint Cycles Commuting

  1. Jun 19, 2014 #1
    1. The problem statement, all variables and given/known data
    I'm taking an online class this summer and the notes gave the proposition that Disjoint Cycles Commute with the following proof.

    Proof. Let σ and τ represent two disjoint cycles in Sn and choose some arbitrary
    j ∈ {1,2, . . . , n}. Since σ and τ are disjoint, at most one of them fixes j, so suppose τ
    fixes j. But then τ must also fix σ.j since the cycles are disjoint. Hence στ.j = σ.k and
    τσ.j = σ.j.

    What does this proof mean when it says that it "fixes" j? what exactly does commute mean again? can anyone help me make sense of this?

    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jun 19, 2014 #2

    pasmith

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    Homework Helper

    [itex]\sigma[/itex] fixes [itex]j[/itex] if and only if [itex]\sigma(j) =j[/itex].

    [itex]\sigma[/itex] and [itex]\tau[/itex] commute if and only if for all [itex]j \in \{1, 2, \dots, n\}[/itex], [itex]\sigma(\tau(j)) = \tau(\sigma(j))[/itex].
     
  4. Jun 20, 2014 #3
    At most or at least?
     
  5. Jun 20, 2014 #4
    The notes say at most, but there have been errors before, i'll bring it up with him at our next meeting. Thanks guys :D
     
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