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Proof by Induction?

  1. May 24, 2017 #1
    1. The problem statement, all variables and given/known data
    Consider the symmetric group ##S_n##. I am trying to establish that ##(i,i+1)=(1,2,...,n)(i-1,i)(1,2,...,n)^{-1}##

    2. Relevant equations


    3. The attempt at a solution

    I am trying to decide whether I need double induction or not. I have done several calculations to see whether I can get away with one induction, but it isn't clear to me whether this is possible. I could use some hints.
     
  2. jcsd
  3. May 24, 2017 #2

    fresh_42

    Staff: Mentor

    Why don't you simply apply the mappings for the cases ##j< i-2\, , \,j > i-1## and ##j \in \{i-2,i-1\}## if read from left to right? It is no recursive definition but an explicit formula, so why use induction?
     
  4. May 24, 2017 #3
    Hold on! I forgot that I proved that ##\sigma (x_1,x_2,...,x_n) \sigma^{-1} = (\sigma(x_1),...,\sigma(x_n))##, which makes the problem trivial.
     
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