Let $G$ be a group of order $n$ and its subgroup lattice be order-isomorphic to that of $\Bbb Z\_n$. Is $G$ cyclic‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌?



a characterization for cyclic groups
Yes, finite cyclic groups are exactly the finite groups whose lattices of subgroups are distributive. The lattice of subgroups of $\mathbb{Z}/n$ is isomorphic to the dual of a divisibility lattice (which is distributive).


