1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted