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Symmetric group question

  1. Dec 4, 2011 #1
    1. The problem statement, all variables and given/known data

    [itex]sgn(\sigma)=0\iff\sigma[/itex] is not a bijection.

    3. The attempt at a solution

    [itex](\rightarrow)[/itex]Let [itex]sgn(\sigma)=0[/itex]. Then, [itex]\Pi_{1\leq i<j\leq n}\frac{\sigma(j)-\sigma(i)}{j-i}=0. [/itex]. For some [itex]i[/itex] and [itex]j[/itex], [itex]i\neq j [/itex], [itex]\sigma(i)=\sigma(j) [/itex]. Thus, [itex]\sigma[/itex] is not an injection.

    [itex](\leftarrow)[/itex] [itex]\sigma[/itex] not a bijection [itex]\rightarrow[/itex] it is not an injection, same argument as above, [itex]sgn(\sigma)=0[/itex].

    I think I went wrong somewhere. Any ideas?
     
    Last edited: Dec 4, 2011
  2. jcsd
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