# Divisibility-closedness is not finite-join-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 join of subgroups is not a divisibility-closed subgroup of .

## Related facts

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

## Proof

Suppose is the generalized dihedral group corresponding to the additive group of rational numbers. Let and both be subgroups of order two generated by different reflections. Then, the following are true:

- is divisible by all primes other than 2.
- and are both divisibility-closed subgroups on account of being finite groups.
- is isomorphic to the infinite dihedral group. It is not divisible by any primes, and in particular it is not divisibility-closed in .