Even and Odd Permutations as Products of Transpositions

# Even and Odd Permutations as Products of Transpositions

Consider the finite $n$-element set $\{ 1, 2, ..., n \}$. On the Decomposition of Permutations as Products of Transpositions page we noted that a permutation need not have a distinct decomposition into a product of transpositions. We also noted that the number of transpositions need not be unique. That said, one nice feature about the set $S_n$ of permutations of the elements in $\{ 1, 2, ..., n \}$ is that they can be categorized into two distinct sets - namely those permutations that can be written as the product of an even number of transpositions and those permutations that can be written as the product of an odd number of transpositions.

 Definition: A permutation $\sigma$ of the finite $n$-element set $\{ 1, 2, ..., n \}$ is said to be Even if $\sigma$ can be written as a product of an even number of transpositions and is said to be Odd if $\sigma$ can be written as a product of an odd number of transpositions.

For example, consider the the set $\{ 1, 2, 3, 4 \}$ and the following permutation $\sigma$:

(1)
\begin{align} \quad \sigma = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 3 & 1 & 2 & 4 \end{pmatrix} \end{align}

Notice that $\sigma = (132) = (12) \circ (13)$. Therefore since $\sigma$ can be written as a product of $2$ (an even number) transpositions, we have that $\sigma$ is an even permutation.

From the definition of even/odd permutations above, it is not clear whether $\epsilon$ is an even or an odd permutation. Furthermore, it's not clear whether or not a permutation can be both even and odd. On The Identity Permutation page we will prove that the identity permutation $\epsilon$ of elements in $\{ 1, 2, ..., n \}$ is even and that a permutation cannot be both even and odd.