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. $\square$
Best Answer
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.