Exercise
Write (1,2,3,4,5)(2,3,4,5,6) as a product of cycles of even length.
(3,1)(3,5,6,1)(1,6)(1,2,4,6).
(3,5,6)(1,2,4).
(1,2)(3,6)(1,4)(3,5)
This is true.
These cycles have odd length.
This product of transpositions defines the permutation (3,5,6)(1,4,2).