Consider Sn for n ≥ 2 and let σ be a fixed odd permutation. Show that every odd permutation in Sn is a product of σ and a permutation in An
Let be an odd permutation in . We must show that there exists an even permutation such that . Indeed, we may take , since as the product of two odd permutations, it is an even permutation, and
For completeness, let’s prove directly that is even. From the definition of an odd permutation, there exist a finite number of transpositions for some odd such that
Similarly, since is also an odd permutation, there exist a finite number of transpositions for some odd such that . Consider now the permutation
. This lies in . Indeed we have
The sum of two odd numbers is even, and so it follows that this is an even permutation.