Can we swap items in a list an odd number of times without changing it?

Question

Suppose there is a list with finitely many distinct items. In each move we swap two of them. How to show that it is impossible to make moves odd times and make the list back to the original state? (Or is it actually possible?)

Answer 1

Every permutation can be expressed as a composition of transpositions, which are permutations that just switch two elements. Perhaps the fundamental theorem to intro permutations is that the number of transpositions for a given permutation is not fixed, but the parity of the number is. That is, if a permutation can be achieved from an even number of transpositions, then it cannot be achieved from an odd number of transpositions, and vice-versa.

With that under our belt, the identity permutation is easily seen to be even. Therefore, an odd number of transpositions will never get a list back to its original state.