Difference between revisions of "Powering-invariance is not finite-join-closed"

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It is possible to have a [[group]] <math>G</math> and [[subgroup]]s <math>H</math> and <math>K</math> of <math>G</matH> such that both <math>H</math> and <math>K</math> are both [[powering-invariant subgroup]]s of <math>G</math> but the [[join of subgroups]] <math>\langle H, K \rangle</math> is not a powering-invariant subgroup of <math>G</math>.
 
It is possible to have a [[group]] <math>G</math> and [[subgroup]]s <math>H</math> and <math>K</math> of <math>G</matH> such that both <math>H</math> and <math>K</math> are both [[powering-invariant subgroup]]s of <math>G</math> but the [[join of subgroups]] <math>\langle H, K \rangle</math> is not a powering-invariant subgroup of <math>G</math>.
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==Related facts==
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* [[Powering-invariance is strongly intersection-closed]]
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* [[Divisibility-closedness is not finite-join-closed]]
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* [[Divisibility-closedness is not finite-intersection-closed]]
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===Nilpotent case===
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* [[Powering-invariance is strongly join-closed in nilpotent group]]
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* [[Divisibility-closedness is strongly join-closed in nilpotent group]]
  
 
==Proof==
 
==Proof==
 
===Non-abelian example===
 
  
 
Suppose <math>G</math> is the [[generalized dihedral group]] corresponding to the additive [[group of rational numbers]]. Let <math>H</math> and <math>K</math> both be subgroups of order two generated by different reflections. Then, the following are true:
 
Suppose <math>G</math> is the [[generalized dihedral group]] corresponding to the additive [[group of rational numbers]]. Let <math>H</math> and <math>K</math> both be subgroups of order two generated by different reflections. Then, the following are true:
  
 
* <math>G</math> is powered over all primes other than 2.
 
* <math>G</math> is powered over all primes other than 2.
* <math>H</math> and <math>K</math> are both powering-invariant subgroups on account of being finite groups.
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* <math>H</math> and <math>K</math> are both powering-invariant subgroups on account of being finite subgroups (see [[finite implies powering-invariant]]).
 
* <math>\langle H, K \rangle</math> is isomorphic to the [[infinite dihedral group]]. It is not powered over any primes, and in particular it is not powering-invariant in <math>G</math>.
 
* <math>\langle H, K \rangle</math> is isomorphic to the [[infinite dihedral group]]. It is not powered over any primes, and in particular it is not powering-invariant in <math>G</math>.

Latest revision as of 21:08, 31 March 2013

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|

Statement

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.

Related facts

Nilpotent case

Proof

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 subgroups (see finite implies powering-invariant).
  • \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.