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.

##### Definition6.3.2

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.

##### Example6.3.5

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.

##### Example6.3.6

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.

##### Definition6.3.8

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].