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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?

Shakira S
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2 Answers2

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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?

I hope this helps ^_^

Chris Grossack
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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 $(24)$.

  • okay so ive attempted and so far, I have (13)=(12)(23)(12) and (14)=(13)(34)(13) ? not too confident though – Shakira S Feb 25 '21 at 00:31
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    To finish, note that $(24)=(12)(14)(12)$. –  Feb 25 '21 at 00:42
  • so If I explain this correctly with the transpositions found, it can be said that all are conjugate like you mentioned thus if we were to form a matrix it would generate all elements of S4? – Shakira S Feb 25 '21 at 00:53
  • Well, as was pointed out in the other answer, all the permutations can be written as a product of transpositions. I just mentioned conjugacy to help you along. –  Feb 25 '21 at 01:22