Powering-invariance is not finite-join-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., powering-invariant subgroup) not satisfying a subgroup metaproperty (i.e., finite-join-closed group property).
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- Powering-invariance is strongly intersection-closed
- Divisibility-closedness is not finite-join-closed
- Divisibility-closedness is not finite-intersection-closed
- Powering-invariance is strongly join-closed in nilpotent group
- Divisibility-closedness is strongly join-closed in nilpotent group
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 powered over all primes other than 2.
- and are both powering-invariant subgroups on account of being finite groups.
- is isomorphic to the infinite dihedral group. It is not powered over any primes, and in particular it is not powering-invariant in .