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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3Answered in http://mathoverflow.net/questions/35455/does-subgroup-structure-of-a-finite-group-characterize-isomorphism-type (see Tony Hyunh's answer). – Igor Rivin May 19 '14 at 18:38
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1Ore's theorem: a group is locally cyclic iff its subgroups lattice is distributive. See a proof here. – Sebastien Palcoux Sep 24 '14 at 14:45
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this question is somewhat old. thanks for the link. – Minimus Heximus Sep 24 '14 at 14:48
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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).
Dietrich Burde
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btw, it seems to be the divisibility lattice itself and not its dual. – Minimus Heximus May 19 '14 at 19:05
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