A note on groups with self-normalizing cyclic subgroups

In this article, we prove that if all non-trivial cyclic subgroups of a group $G$ are self normalizing and $G$ satisfies the implication $$o(x)\mathrm{\ne }o(y)\Rightarrow o(xy)\mathrm{\ne }o(x),o(y),$$ for all non-trivial elements $x$ and $y$, then $G$ is a $p$-group or simple.