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