Answer to Question #178276 in Abstract Algebra for 123

Question

Consider S_n for n ≥ 2 and let σ be a fixed odd permutation. Show that every odd permutation in S_n is a product of σ and a permutation in A_n

.

Answer 1

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.