Permutation of set Suppose I have a set which contain finite elements from $1,2,\dots,n$, where $n$ is odd. Let there are two permutations $(a_1,a_2,\dots,a_n)$ and $(b_1,b_2,b_3,\dots,b_n)$ defined on this set. Now I have to prove that $$(a_1-b_1)^2(a_2-b_2)^2(a_3-b_3)^2\cdots(a_n-b_n)^2$$ is a even number.
My findings/questions are as follows :-

*

*Every permutation of a set is either a cycle or product of disjoint cycles, so these to permutations are either be cycle or product of disjoint cycles.

*Can the product of square terms can be a cycle of the set?

*Or the given product of square numbers can be converted onto any one of permutations, so that can claim to even or odd.

Please clear it will be helpful for me

Since $n$ is odd, one of the factors $a_i-b_i$ is even, where $a_i,b_i$ are odd. Then the whole product is even.
