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

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SUMMARY

The symmetric group S4 can be generated by the elements (12) and (143), where (12) is a 2-cycle and (143) is a 3-cycle. The relationships established include x^2 = e, y^3 = e, and (xy)^4 = e, confirming that these elements satisfy the generating conditions for S4. The discussion references the theorem that S_n can be generated by transpositions and cycles, validating the proposed generators. The calculations demonstrate that (12)(143) yields a 4-cycle, reinforcing the conclusion that these elements generate the entire group.

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Homework Statement


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##


Homework Equations





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
 
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