Permutation as a Product of Transposition

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Every permutation in S_n for n>1 can be expressed as a product of two cycles. To prove this, one can start by writing the permutation as a product of cycles and then demonstrate how each cycle can be represented as a product of transpositions. An example provided is the cycle (1234), which can be expressed as (14)(13)(12). The confusion arises around the derivation of the formula for transpositions, specifically the sequence (a_1, a_k)(a_1, a_k-1)...(a_1, a_2). Clarification on this derivation is sought to solidify understanding of the theorem.
liger123
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hi guys.. can you help me prove this theorem?
Every permutation S_n where n>1 is a product of 2 cycles..
i got a little confused with some books' proof..thnx
 
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Can you show some work you've done on this?

The proof that comes to mind for me is to write each permutation as a product of cycles (you know how to do this, right?), and then you can explicitly describe how each cycle is a product of transpositions. For example, (1234) = (14)(13)(12).
 
Yes, the thing that i don't understand is how the formula was derived. this is the formula
(a_1, a_k) (a_1, a_k-1) ... (a_1, a_2).
 

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