Is it true that $a$ and $b$ should be disjoint permutations?

Question

Let $a,b \in S_n$ and $ab=ba$ and $b$ moves some points that not moved by $a$. Is it true that $a$ and $b$ should be disjoint permutations?

EDIT: We can consider $b=a^kc$ where $a$ and $c$ are disjoint, to construct a counterexample class. Do you know other constructions?

Answer 1

No. Take $a=(1,2)$ and $b=(1,2)(3,4)$.

Answer 2

Not sure what you mean by ``disjoint permutation", but maybe $a=(1,2)$ and $b=(1,2)(3,4)$ (in $S_4$ for example) is a counterexample?

Answer 3

Consider a cycle $\,a\,$ and $\,b\,$ a product of $\,a\,$ with a cycle disjoint from it.