Let $G$ be a group acting on a set $\Omega$ and let $p$ be a prime. Suppose that for each $\alpha \in\Omega$ there is a $p$-element $x \in G$ such that $\alpha$ is the only point fixed by $x$. If $\Omega$ is finite, show that $G$ is transitive on $\Omega$; and if $\Omega$ is infinite, show that $G$ has no finite orbit on $\Omega$.
Any hints how to solve this problem?
EDIT: My initial thoughts were all wrong as I am assumed uniqueness of the $p$-element there.