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

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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).

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  • $\begingroup$ btw, it seems to be the divisibility lattice itself and not its dual. $\endgroup$ – user47958 May 19 '14 at 19:05
  • $\begingroup$ @MinimusHeximus: The divisibility lattice is self-dual. $\endgroup$ – Emil Jeřábek supports Monica May 19 '14 at 21:10

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