Permutation as a Product of Transposition

[liger123]

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

 

[adriank]

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).

 

[liger123]

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).