How to show that cycles of even degree are odd permutations? What I understand is that cycles are actually referring to cyclic groups (if not, please correct me). So we have some cyclic group $\langle g \rangle$ such that $|\langle g \rangle | = n\;mod\;2 = 0$. If given a set $\Omega$ with $g\in G\in Sym(\Omega)$, show that $\langle g \rangle$ can be written as a product of an odd number of transpositions.
$(1\ 2\ \dots n) = (1\ n)(1\ n-1)\cdots(1\ 2)$
This shows an $n$-cycle can be written as (a product of) $n-1$ transpositions (just replace $k$ with $a_k$).