Does every non prime order finite group have at least one non trivial proper cyclic subgroup?
Let $a\neq e$ be an element of $G$. Then $\langle a \rangle$ is always a subgroup of $G$, rather it is a cyclic subgroup of $G$. But my doubt arises, what will guarantee us that we shall always have at least one such proper subgroups generated by the elements of $G.$