# Rewriting a cycle as a product of tranpositions

I'm stuck on a homework problem that asks

Write the cycle $(2 3 4)$ as a product of transpositions in $\{(1 2),(1 3), (1 4)\}$

I'm fine with computing permutations, but I don't know what the problem is asking. Specifically, what does the last part of the question imply?

• It means: "Write the permutation $(234)$ as a product of transpositions, with the added restriction that you can use only the transpositions $(12)$, $(13)$ and $(14)$." You cannot use $(23)$ directly, for instance. – Srivatsan Dec 7 '11 at 21:47
• OK - that makes sense... Trying with guess and check...would it be (14)(13)(12)? – pigishpig Dec 7 '11 at 21:51
• @pigishpig, well where does 1 go in (234), and in your product? – Alex Dec 7 '11 at 21:55
• I am very glad if you can help me on this. Thank you. math.stackexchange.com/questions/423297/… – Maizon Jun 18 '13 at 2:58

Consider $(12)(13)$. 2 goes to 3, which is good, but 3 goes to 1, which is bad.
So consider $(12)(13)(14)$. Now 2 still goes to 3, and 3 now goes to 4, which is good, but 4 goes to 1, which is bad.
So consider $(12)(13)(14)(12)$.