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October 22nd, 2007, 04:53 PM  #1 
Newbie Joined: Oct 2007 Posts: 3 Thanks: 0  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^n1.

October 23rd, 2007, 12:22 AM  #2 
Site Founder Joined: Nov 2006 From: France Posts: 824 Thanks: 7 
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)^(j2+n)(1,2)(1,...,n)^(ni+1), which solves the problem (provided I didnt do any computational mistake, but the method is correct anyhow).


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factors, permutation, product 
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