Divisibility-closedness is not finite-intersection-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., divisibility-closed subgroup) not satisfying a subgroup metaproperty (i.e., finite-intersection-closed subgroup property).
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In fact, we can choose our example so that is a divisible abelian group, and hence, so are and .
- Powering-invariance is strongly intersection-closed
- Divisibility-closedness is not finite-join-closed
- Powering-invariance is not finite-join-closed
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 .