Divisibility-closedness is not finite-join-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).
View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about divisibility-closed subgroup|Get more facts about finite-intersection-closed subgroup property|
- Powering-invariance is not finite-join-closed
- Divisibility-closedness is not finite-intersection-closed
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 .