# 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>. | ||

+ | |||

+ | ==Related facts== | ||

+ | |||

+ | * [[Powering-invariance is strongly intersection-closed]] | ||

+ | * [[Divisibility-closedness is not finite-join-closed]] | ||

+ | * [[Divisibility-closedness is not finite-intersection-closed]] | ||

+ | |||

+ | ===Nilpotent case=== | ||

+ | |||

+ | * [[Powering-invariance is strongly join-closed in nilpotent group]] | ||

+ | * [[Divisibility-closedness is strongly join-closed in nilpotent group]] | ||

==Proof== | ==Proof== | ||

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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 | + | * <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)notsatisfying 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 and subgroups and of such that both and are both powering-invariant subgroups of but the join of subgroups is not a powering-invariant subgroup of .

## Related facts

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

### Nilpotent case

- Powering-invariance is strongly join-closed in nilpotent group
- Divisibility-closedness is strongly join-closed in nilpotent group

## 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 powered over all primes other than 2.
- and are both powering-invariant subgroups on account of being finite subgroups (see finite implies powering-invariant).
- is isomorphic to the infinite dihedral group. It is not powered over any primes, and in particular it is not powering-invariant in .