show a subset generates a symmetric group

Question

How would I show that the subset $V = \{(1,2), (2,3), (3,4)\}$ generates the symmetric group $S_4$?

I am just being introduced to group theory this is all new. I have unsuccessfully attempted to solve this question. . How do you prove this?

Answer 1

Welcome to MSE!

Here is a hint:

We know that every permutation can be written as a product of transpositions (see here, for instance). So if you can get every transposition, then you can get every permutation (do you see why?).

Of course, you already have $3$ transpositions available! Can you combine these to get the others?

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I hope this helps ^_^

Answer 2

You can get all the transpositions from those three. There are ${4\choose2}=6$ transpositions total, so there are sort of $3$ things to check.

For instance, $(12)(23)(12)=(13)$.

(One more hint: the conjugation action is transitive; that is all the transpositions are conjugate)

So it remains to find $(14)$ and $(24)$.