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Statement:
No permutation in $S_n$ can be expressed both as a product of an even number of transpositions and as a product of an odd number of transpositions.
Proof:
Note $S_A\simeq S_B$, if $|A|=|B|$. We work with permutations of the n rows of the $n\times n$ identity matrix $I_n$, rather than of the numbers $1,2,\dotsb n$.The identity matrix has determinant 1. Interchanging any two rows of a square changes the sigh of the determinant. Let $C$ be a matrix obtained by a permutation $\sigma$ of the rows of $I_n$. If $C$ could be obtained from $I_n$ by both an even number and an odd number of transpositions of rows, its determinant would have to be both 1 and -1, which is impossible. Thus $\sigma$ cannot be expressed both as product of an even number and an odd number of transpositions.