 Groups with no proper nontrivial subgroups

A nontrivial group has no proper nontrivial subgroups exactly when it is a cyclic group of prime group order. To prove necessity without presupposing finiteness, take $g\ne e$. Its cyclic subgroup is nontrivial and hence equals $G$. An infinite-order generator would give the proper nontrivial subgroup $⟨g^2⟩$, so its order $N$ is finite. If $N$ were composite, a proper divisor $d$ with $1<d<N$ would make $⟨g^d⟩$ another proper nontrivial subgroup. Thus $N$ is prime. Conversely Lagrange's theorem excludes intermediate subgroup orders in a prime-order group. The qualification “nontrivial” is necessary because every group with more than one element has the proper trivial subgroup $\{e\}$.