Commutativity of special type of permutations

Question

Let $p_1,p_2$ be permutations on the set $S=\{1,2,...,n \}$ and define $U_1:=\{k\in S:p_1(k)\ne k\}$ , similarly we can define $U_2$ . Now I can prove that if $U_1 \cap U_2=\phi$ , then $p_1p_2=p_2p_1$ ; so I ask is the converse true i.e. if $p_1p_2=p_2p_1$ then is it true that $U_1 \cap U_2=\phi$ ?

Answer 1

Take $S=\{1,2\}$, $p_1=p_2=(1,2)$. Now $p_1p_2=p_2p_1=(1)$ and $U_1=U_2=S$.

If you didn't like $p_1=p_2$, take $S=\{1,2,3\}$, $p_1=(1,2,3)$ and $p_2=(3,1,2)=p_1^{-1}$. See that $p_1p_2=p_2p_1=(1)$ but $U_1=U_2=S$.

Answer 2

Certainly not: If $p_1 = p_2$, then $p_1 p_2 = p_2 p_1$ but $U_1 = U_2$, so $U_1 \cap U_2 \neq \emptyset$ unless $p_1$ is trivial.