# Difference between revisions of "Contranormality is upper join-closed"

From Groupprops

(New page: {{subgroup metaproperty satisfaction| property = contranormal subgroup| metaproperty = upper join-closed subgroup property}} ==Statement== ===Statement with symbols=== Suppose <math>H \...) |
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<math>H \le K</math> is a contranormal subgroup if for any <math>L \le K</math> containing <math>H</math> such that <math>L</math> is normal in <math>K</math>, <math>L = K</math>. | <math>H \le K</math> is a contranormal subgroup if for any <math>L \le K</math> containing <math>H</math> such that <math>L</math> is normal in <math>K</math>, <math>L = K</math>. | ||

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+ | ==Related facts== | ||

+ | |||

+ | ===Stronger facts=== | ||

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+ | * [[Contranormality is UL-join-closed]] | ||

+ | ===Applications=== | ||

+ | |||

+ | * [[Paranormal implies polynormal]] | ||

==Facts used== | ==Facts used== | ||

− | # [[Normality satisfies transfer condition]]: If <math>L \triangleleft G</math> is a normal subgroup, and <math>K \le G</math>, then <math>L \cap K</math> is normal in <math>K</math>. | + | # [[uses::Normality satisfies transfer condition]]: If <math>L \triangleleft G</math> is a normal subgroup, and <math>K \le G</math>, then <math>L \cap K</math> is normal in <math>K</math>. |

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

## Latest revision as of 12:58, 25 September 2008

This article gives the statement, and possibly proof, of a subgroup property (i.e., contranormal subgroup) satisfying a subgroup metaproperty (i.e., upper join-closed subgroup property)

View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties

Get more facts about contranormal subgroup |Get facts that use property satisfaction of contranormal subgroup | Get facts that use property satisfaction of contranormal subgroup|Get more facts about upper join-closed subgroup property

## Contents

## Statement

### Statement with symbols

Suppose is a subgroup, and , is an indexed family of subgroups with for each . Then, if is contranormal in each , is also contranormal in the [join of subgroups|join]] of the s.

## Definitions used

### Contranormal subgroup

`Further information: contranormal subgroup`

is a contranormal subgroup if for any containing such that is normal in , .

## Related facts

### Stronger facts

### Applications

## Facts used

- Normality satisfies transfer condition: If is a normal subgroup, and , then is normal in .

## Proof

**Given**: , family of subgroups with , and contranormal in each .

**To prove**: is normal in the join of all the s.

**Proof**: Suppose is a normal subgroup of the join of the s, containing . Then, for each , is a subgroup of containing , and by fact (1), it is normal in . Since is contranormal in , for each , so for each . Thus, must equal the join of the s.