Three even-length cycles make this permutation odd.Suggest a slogan
The permutation \(\sigma=(1\,2\,3\,4\,5\,6)(7\,8\,9)(10\,11)(12\,13\,14\,15)(16\,17\,18)\in S_{18}\) has sign \(\varepsilon(\sigma)=-1\).
A cycle of length \(\ell\) is a product of \(\ell-1\) transpositions, so its sign is \((-1)^{\ell-1}\) (FD-GHDF2). The disjoint cycles of \(\sigma\) have lengths \(6,3,2,4,3\), so \[ \varepsilon(\sigma)=(-1)^{5+2+1+3+2}=(-1)^{13}=-1. \] Equivalently, \(\sigma\) has three cycles of even length.
What is the sign of the permutation $\sigma=(123456)(789)(10~11)(12~13~14~15)(16~17~18)$?