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

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In summary, the symmetric group on 4 letters, S4, can be generated by two elements x and y such that x^2 = y^3 = (xy)^4. The elements (12) and (143) can be used to generate S4, as shown by the theorems in the text. This is because (13)(14) = (143) and (1234) = (143)(143)(12), meaning that they both generate the same group. Additionally, (12)(143) = (1432), which is a 4-cycle that is the identity when raised to the fourth power. While not written rigorously, this demonstrates the correct mindset.
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lus1450
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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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  • #2
It looks good to me.
 

1. What is S4?

S4 is the symmetric group on four elements, which consists of all possible permutations of four distinct objects.

2. What is a 2-cycle and a 3-cycle?

A 2-cycle is a permutation that switches the positions of two elements, while leaving all other elements unchanged. A 3-cycle is a permutation that switches the positions of three elements, while leaving all other elements unchanged.

3. How can a 2-cycle and a 3-cycle generate S4?

By composing a 2-cycle and a 3-cycle, we can create all possible permutations of four elements. This is because any permutation can be expressed as a composition of disjoint cycles, and a 2-cycle and a 3-cycle can generate all possible disjoint cycles of length 2 and 3, respectively.

4. Can S4 be generated by other types of cycles?

Yes, S4 can also be generated by a pair of 4-cycles or a pair of 2-cycles and a 4-cycle. However, the 2-cycle and 3-cycle combination is the most commonly used to prove that S4 is generated by a small number of elements.

5. Why is proving that S4 is generated by a 2-cycle and a 3-cycle important?

Proving this fact is important because it helps us understand the structure of the symmetric group and the relationship between different types of cycles. It also has applications in various areas of mathematics, such as group theory and abstract algebra.

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