# Divisibility-closedness is not finite-intersection-closed

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., divisibility-closed subgroup)notsatisfying a subgroup metaproperty (i.e., finite-intersection-closed subgroup property).

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## Statement

It is possible to have a group and subgroups of such that both and are divisibility-closed subgroups of but the intersection of subgroups is not.

In fact, we can choose our example so that is a divisible abelian group, and hence, so are and .

## Related facts

- Powering-invariance is strongly intersection-closed
- Divisibility-closedness is not finite-join-closed
- Powering-invariance is not finite-join-closed

## Proof

Let be the external direct product of the (additive) group of rational numbers and the group of rational numbers modulo integers . In other words, we have:

Consider the subgroups and defined as follows:

- is the first direct factor.
- Let be the natural quotient map. Define:

.

Now, the following are true:

- is a divisible abelian group (i.e., -divisible for all natural numbers ), because both its direct factors are.
- and are both isomorphic to , hence are both divisible abelian groups (i.e., -divisible for all ), and hence, are divisibility-closed in .
- The intersection is the subgroup:

This is isomorphic to the group of integers, which is not divisible by any . Hence, it is not divisibility-closed in .