# Permutations $\{1, …, n\}$ with all cycles even

I am trying to solve a problem that involves permutations of $\{1, ..., n\}$ with all cycles even. What does this mean? Could you please give an example of such permutation?

I understand that, e.g. when $n = 4$, a permutation $\langle2,1,4,3\rangle$ involves 2 cycles, namely $(1,2)$ and $(3,4)$. Are these "even cycles"?

• I do not want to mark this as a duplicate, but I think you can get an answer here. What does it take to convert an odd permutation to an even permutation ? – Shailesh Sep 23 '15 at 1:17
• This means a permutation that can be written as a product of cycles of even length (i.e. of signature $+1$). Your example illustrates this correctly. – Alex M. Sep 23 '15 at 21:09
• @AlexM. Let $p$ be an even permutation. If $p$ is a cycle, then $p$ has odd length, right? – user198044 Oct 10 '18 at 13:43
• @JackBauer: Rereading my comment above, it should have been "number of inversions" instead of "length". And yes, you are right. – Alex M. Oct 10 '18 at 14:01
• @AlexM. Thank you! – user198044 Oct 10 '18 at 14:05