# How to show that cycles of even degree are odd permutations?

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.

• How about we say cycles of even length rather than even "degree"? This accords with terminology for permutations. – hardmath Oct 1 '15 at 0:55
• @hardmath, be my guest, but I just used the exact terminology from my book. – bourbaki4481472 Oct 1 '15 at 0:59
• Fair enough, but I'm wondering why you do not rely on the definition your book uses for a cycle permutation and the classification of cycles as even or odd degree? – hardmath Oct 1 '15 at 1:06
• @hardmath, because this is from an introduction section of a short book so the author didn't go through all the definitions carefully. – bourbaki4481472 Oct 1 '15 at 2:34

$(1\ 2\ \dots n) = (1\ n)(1\ n-1)\cdots(1\ 2)$
This shows an $n$-cycle can be written as (a product of) $n-1$ transpositions (just replace $k$ with $a_k$).