Parity of permutation If I know that the parity of a permutation is the parity of the number of transpositions, how can prove that the parity of the permutation is the parity of permutation decrement? Permutation decrement is the difference between the number of truly movable elements and the number of independent cycles

I'm going to assume you're trying to prove that the parity of a $k$-cycle is equal to the parity of $k-1$. If this is the case, note that $$(a_1~~a_2~~\cdots~~a_{k-1}~~a_k) = (a_1~~a_k)(a_1~~a_{k-1}) \cdots(a_3~~a_1)(a_2~~a_1)$$ The expression on the right-hand side is a product of $k-1$ transpositions.