Powering-invariance is not finite-join-closed

From Groupprops
Revision as of 01:13, 17 February 2013 by Vipul (talk | contribs) (Created page with "{{subgroup metaproperty dissatisfaction| property = powering-invariant subgroup| metaproperty = finite-join-closed group property}} ==Statement== It is possible to have a ...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
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).
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 powering-invariant subgroup|Get more facts about finite-join-closed group property|


It is possible to have a group G and subgroups H and K of G such that both H and K are both powering-invariant subgroups of G but the join of subgroups \langle H, K \rangle is not a powering-invariant subgroup of G.


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

  • G is powered over all primes other than 2.
  • H and K are both powering-invariant subgroups on account of being finite groups.
  • \langle H, K \rangle is isomorphic to the infinite dihedral group. It is not powered over any primes, and in particular it is not powering-invariant in G.