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I know that if I have any permutation, this permutation can be written as the product of transpositions. Now the number of these transpositions may be odd or even depending on my permutation. I also know that any transposition can be written as the product of simple transpositions (the pair $(i,i+1)$). So this means that any permutation can be written as the product of simple transpositions and the number of these transpositions may be odd or even depending on the permutation. Have I missed anything?

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So what exactly is your question? –  Alexander Thumm Mar 22 '11 at 9:02
    
that my statement is correct or not? –  Vafa Khalighi Mar 22 '11 at 9:04
    
No, you haven't missed anything, and yes, your statement is correct :-) –  joriki Mar 22 '11 at 9:07
    
You may have a look here: en.wikipedia.org/wiki/Parity_of_a_permutation –  anonymous Mar 22 '11 at 9:07
    
Please help me. Thank you. math.stackexchange.com/questions/423297/… –  Maizon Jun 18 '13 at 2:51

1 Answer 1

No, you have not missed anything. But you should probably state explicitly that no permutation can be written both as an even number of transpositions and an odd number of transpositions.

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