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Proving S4 is generated by a 2-cycle and a 3-cycle

  1. Dec 8, 2013 #1
    1. The problem statement, all variables and given/known data
    Show ##S_4## (symmetric group on ##4## letters) can be generated by two elements ##x## and ##y## such that ##x^2 = y^3 = (xy)^4##


    2. Relevant equations



    3. The attempt at a solution
    I'm guessing I can use ##(12)## and ##(143)##. I got this since I know ##S_n = \langle (12), (13), (14) \rangle## by theorem in my text, and ##(13)(14) = (143)##. I know by another theorem that ##S_n = \langle (12),(1234) \rangle \text{ and } (1234) = (143)(143)(12)##, so I believe that means I am allowed to say they both generate the same group. Furthermore, ##(12)(143) = (1432)##, which is a ##4##-cycle that will be the identity when raised to the fourth. I know here it's not written rigorously, but that is my mindset. Is this correct? Thanks in advance
     
  2. jcsd
  3. Dec 9, 2013 #2
    It looks good to me.
     
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