In group (S10,o) the decomposition of a permutation p into disjoint cycles is p=(134)(2867). The order is found using the least common multiple (LCM) of cycle lengths, in this case, LCM(3,4) = 12. So, the order is 12, and the transpositions are (13)(14)(28)(26)(27), making it an odd number.
Now, I need to find all permutations with the same order but different parity. To obtain an order of 12, I can multiply cycles of lengths 4,3 or 2,2,3. It seems that 4,3 always results in an odd parity, while 2,2,3 always results in an even parity due to the number of transpositions.
However, I am aware that changing the order of two elements in permutations should change the parity. For instance, (143)(2867) is expected to be even (as I changed the order of 2 elements), but it's observed to be odd (having 5 transpositions). Can somebody explain where the misconception might be?
Changing in the order of elements in a linear ordering can change the parity of the permutation represented by the linear ordering, but changing the ordering of points inside a cycle in the disjoint-cycle representation of a permutation won't change the parity.