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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?

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    $\begingroup$ 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. $\endgroup$ – Srivatsan Dec 7 '11 at 21:47
  • $\begingroup$ OK - that makes sense... Trying with guess and check...would it be (14)(13)(12)? $\endgroup$ – pigishpig Dec 7 '11 at 21:51
  • $\begingroup$ @pigishpig, well where does 1 go in (234), and in your product? $\endgroup$ – Alex Dec 7 '11 at 21:55
  • $\begingroup$ I am very glad if you can help me on this. Thank you. math.stackexchange.com/questions/423297/… $\endgroup$ – Maizon Jun 18 '13 at 2:58
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We want 2 to go to 3, 3 to go to 4, and 4 to go to 2 (while 1 goes nowhere).

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 $(12)(13)(14)(12)$.

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  • $\begingroup$ Yes. I've been taught using the right-to-left method, so (12)(14)(13)(12) for me. Thanks for the explanation! $\endgroup$ – pigishpig Dec 7 '11 at 22:50

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