##### Theorem6.3.1

Every permutation in \(S_n\) can be written as a product of transpositions.

Note that every \(k\)-cycle \((a_1a_2\ldots a_k)\in S_n\) can be written as a product of (not necessarily disjoint) transpositions:

\begin{equation*} (a_1a_2\ldots a_k)=(a_1a_k)(a_1a_{k-1})\cdots(a_1a_3)(a_1a_2). \end{equation*}We therefore have the following theorem.

Every permutation in \(S_n\) can be written as a product of transpositions.

We say that a permutation in \(S_n\) is *even* [resp., *odd*] if it can be written as a product of an even [resp., odd] number of transpositions.

Every permutation in \(S_n\) is even or odd, but not both.

For each \(2\leq k\leq n\text{,}\) then a \(k\)-cycle is even if \(k\) is odd, and odd if \(k\) is even.

The proof of this is left as an exercise for the reader.

In \(S_3\text{,}\) the permutations \(e\text{,}\) \((123)=(13)(12)\text{,}\) and \((132)=(12)(13)\) are even, while the permutations \((12)\text{,}\) \((13)\text{,}\) and \((23)\) are odd.

List all of the even [resp., odd] permutations in \(S_4\text{.}\)

We have the following theorem, whose proof is left as an exercise for the reader.

The set of all even permutations in \(S_n\) is a subgroup of \(S_n\text{.}\)

The *alternating group on \(n\) letters* is the subgroup \(A_n\) of \(S_n\) consisting of all of the even permutations in \(S_n\text{.}\)

We end with this theorem, whose proof can be found on p. 93 of [1].

\(|A_n|=(n!)/2\text{.}\)