Finite groups with unique minimal subgroup Let $G$ be a finite group. Let $G$ has a unique non trivial minimal subgroup. Then $G$ is a p-group. How to prove the theorem which says that: If $G$ has a unique non trivial minimal subgroup and if $p$ is not 2, then $G$ is cyclic. I made several attempts but failed eventually. The theorem is false for $p=2$ as shown in one of the answers.

Your "theorem" is false by any interpretation. The quaternion group $Q_8$ is not cyclic, but has a unique minimal non-trivial subgroup $\{1,-1\}$; it is contained in any non-trivial subgroup of$~Q_8$. (And of course if "minimal subgroup" means $\{e\}$, then any non-cyclic group is a counterexample.)
