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October 22nd, 2007, 05:53 PM   #1
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permutation as a product of factors

Prove that all permutions in Sn can be produced as products using only the factors sigma=(1 2) and tau=(1 2 3 ... n). Note that tau^-1= tau^n-1.
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October 23rd, 2007, 01:22 AM   #2
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The set of all permutations in Sn is generated by the set of all transpositions of S_n. It is therefore enough to prove that any transposition can be generated by the elements (permutations) (1,2) and (1,...,n). Well, for any transposition (i,j), we have (i,j)=(1,...,n)^(j-2+n)(1,2)(1,...,n)^(n-i+1), which solves the problem (provided I didnt do any computational mistake, but the method is correct anyhow).
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