Proof verification: Every $3$-cycle is an even permutation.

Can someone please verify whether my proof is correct?

Show that a $3$-cycle is an even permutation.

Proof: Let $\sigma = (a_{1}a_{2}a_{3})$ be a $3$-cycle. Then $\sigma$ can be written as a product of transpositions, with $\sigma = (a_{1}a_{3})(a_{1}a_{2})$. If a permutation is expressed as a product of an even number of transpositions, then it cannot be expressed as an odd number of transpositions. By definition, the $3$-cycle $\sigma$ is even. $\square$

• The long sentence requires a proof, perhaps citing a result already proved. – vadim123 Feb 11 '18 at 20:54
• Is the long sentence even necessary? The definition of an even permutation I know is a product of an even number of transpositions, so just writing $\sigma=(a_1a_3)(a_1a_2)$ would be enough. – Wojowu Feb 11 '18 at 21:00
• @Wojowu That's a good point! However, so is vadim's point - it is deliciously subtle! – user1729 Feb 11 '18 at 21:08
• @vadim123 I included the long sentence to show that it can only be written as even, and so it isn't a specific case that I wrote. Perhaps I do not need to include it then! – user482939 Feb 11 '18 at 23:35

Check that $\sigma = (a_{1}, ..., a_{n}) = (a_{1}, a_{n})(a_{1}, a_{n-1})... (a_{1},a_{2})$ if $n > 1$. Therefore $\sigma$ is even if, and only if, $n$ is odd.
$\textbf{Remember:}$ a permutation $\sigma$ if even when is product of an even number of transpositions.