Subgroups of cyclic groups
Lattice of subgroups
Lattice of subgroups
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In both the finite and the infinite case, the [lattice of subgroups](/wiki/Lattice_of_subgroups "Lattice of subgroups") of a cyclic group is isomorphic to the [dual](/wiki/Duality_%28order_theory%29 "Duality (order theory)") of a [divisibility](/wiki/Divisibility "Divisibility") lattice. In the finite case, the lattice of subgroups of a cyclic group of order *n* is isomorphic to the dual of the lattice of divisors of *n*, with a subgroup of order *n*/*d* for each divisor *d*. The subgroup of order *n*/*d* is a subgroup of the subgroup of order *n*/*e* if and only if *e* is a divisor of *d*. The lattice of subgroups of the infinite cyclic group can be described in the same way, as the dual of the divisibility lattice of all positive integers. If the infinite cyclic group is represented as the additive group on the integers, then the subgroup generated by *d* is a subgroup of the subgroup generated by *e* if and only if *e* is a divisor of *d*.

Divisibility lattices are [distributive lattices](/wiki/Distributive_lattice "Distributive lattice"), and therefore so are the lattices of subgroups of cyclic groups. This provides another alternative characterization of the finite cyclic groups: they are exactly the finite groups whose lattices of subgroups are distributive. More generally, a [finitely generated group](/wiki/Finitely_generated_group "Finitely generated group") is cyclic if and only if its lattice of subgroups is distributive and an arbitrary group is [locally cyclic](/wiki/Locally_cyclic_group "Locally cyclic group") if and only its lattice of subgroups is distributive.. The additive group of the [rational numbers](/wiki/Rational_number "Rational number") provides an example of a group that is locally cyclic, and that has a distributive lattice of subgroups, but that is not itself cyclic.


Is it true that any locally cyclic group has a distributive lattice of subgroups?