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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asked Mar 22, 2011 at 8:55
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1$\begingroup$ So what exactly is your question? $\endgroup$– Alexander ThummMar 22, 2011 at 9:02
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$\begingroup$ that my statement is correct or not? $\endgroup$– Vafa KhalighiMar 22, 2011 at 9:04
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$\begingroup$ No, you haven't missed anything, and yes, your statement is correct :-) $\endgroup$– jorikiMar 22, 2011 at 9:07
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$\begingroup$ You may have a look here: en.wikipedia.org/wiki/Parity_of_a_permutation $\endgroup$– anonymousMar 22, 2011 at 9:07
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$\begingroup$ Please help me. Thank you. math.stackexchange.com/questions/423297/… $\endgroup$– MaizonJun 18, 2013 at 2:51
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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.
answered Mar 22, 2011 at 9:06