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

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Answer 1

Let $\sigma'$ be an odd permutation in $S_n$. We must show that there exists an even permutation $\mu\isin A_n$ such that $\sigma'=\sigma\mu$. Indeed, we may take $\mu=\sigma^{-1}\sigma'$, since as the product of two odd permutations, it is an even permutation, and

$\sigma'=\sigma(\sigma^{-1}\sigma')$

For completeness, let’s prove directly that $\sigma^{-1}\sigma'$  is even. From the definition of an odd permutation, there exist a finite number of transpositions $\tau_1,...,\tau_m$ for some odd $m\isin N$ such that

$\sigma=\tau_1...\tau_m$

Similarly, since $\sigma'$ is also an odd permutation, there exist a finite number of transpositions $\tau'_1,...,\tau'_l$ for some odd $l\isin N$ such that $\sigma'=\tau'_1...\tau'_l$.  Consider now the permutation

$\mu=\sigma^{-1}\sigma'$. This lies in $A_n$. Indeed we have

$\mu=\sigma^{-1}\sigma'=\tau_m...\tau_1\tau'_1...\tau'_l$

The sum of two odd numbers is even, and so it follows that this is an even permutation.