Odd or Even number of transpositions

Question

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?

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.