The symmetric group $S_4$ and cycles as product of permutations

Question

i have a question, i think it is very easy but i didn't see it: I know that:

$$S_4=\langle(1~~2),(2~~3),(3~~4)\rangle$$

Now i want to write $(1~~2~~3)$ and $(1~~2~~3~~4)$ as products of them. How to do that and what is the solution?

Thank you.

Answer 1

If you're familiar with conjugation, then you'll notice it's not to hard to get $(1 n)$ using these generators.

e.g. $(1 3)=(1 2)^{(2 3)}=(2 3)(1 2)(2 3)$

Hence $(1 2 3)=(1 2)(1 3)=(1 2)(2 3)(1 2)(2 3)$

See if you can do the other one by yourself now.