Expressing a cycle/set as odd or even.

Question

I have a cycle here that I have broken down into 2 disjoint cycles, them being (1,5,6) and (2,8). I'm wondering what is the process of telling whether the set is even or odd. Is it the number of disjoint cycles or the size?

Answer 1

A cycle of even length / order is an odd permutation, and vice versa (that's just a conventional dissonance we have to live with; the alternative is worse). The product of two odd or of two even permutations (not just cycles) is even, while the product of one even and one odd permutation (in whatever order) is odd. The identity permutation is considered even.

Armed with this knowledge, we get that the cycle $(1,5,6)$ has length $3$ and is therefore even, and $(2,8)$ has length $2$ and is therefore odd, so their product is odd.