# WNSCDIN implies every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., WNSCDIN-subgroup) must also satisfy the second subgroup property (i.e., subgroup in which every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed)

View all subgroup property implications | View all subgroup property non-implications

Get more facts about WNSCDIN-subgroup|Get more facts about subgroup in which every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed

## Contents

## Statement

If is a WNSCDIN-subgroup of , and is a normalizer-relatively normal conjugation-invariantly relatively normal subgroup of with respect to , then is a weakly closed subgroup of with respect to .

## Definitions used

### WNSCDIN-subgroup

`Further information: WNSCDIN-subgroup`

A subgroup of a group is a WNSCDIN-subgroup if whenever are normal subsets of and is such that , there exists such that .

### Normalizer-relatively normal subgroup

`Further information: Normalizer-relatively normal subgroup`

Suppose . is normalizer-relatively normal in with respect to if is normal in . In other words, .

### Conjugation-invariantly relatively normal subgroup

`Further information: Conjugation-invariantly relatively normal subgroup`

Suppose . is conjugation-invariantly relatively normal in with respect to if is normal in every conjugate of in that contains .

### Weakly closed subgroup

`Further information: Weakly closed subgroup`

Suppose . is weakly closed in relative to if, for any such that , we have .

## Related facts

### Converse

- Weakly closed implies normalizer-relatively normal
- Weakly closed implies conjugation-invariantly relatively normal in finite group

### Applications

- Characteristic subgroup of Sylow subgroup is weakly closed iff it is normal in every Sylow subgroup containing it
- Equivalence of definitions of weakly closed conjugacy functor

## Proof

This proof uses a tabular format for presentation. Provide feedback on tabular proof formats in a survey (opens in new window/tab) | Learn more about tabular proof formats|View all pages on facts with proofs in tabular format

**Given**: Groups such that is a WNSCDIN-subgroup of . is normal in , and is normal in for all such that . We have such that .

**To prove**: (in fact, we'll prove equality).

**Proof**:

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | is a normal subgroup of | Since , conjugating both sides by yields . By the assumption that is normal in every conjugate of containing it, is a normal subgroup of . | |||

2 | is a normal subgroup of . | Step (1) | This follows from the previous step, and conjugating by . | ||

3 | is a normal subgroup of . | normality satisfies intermediate subgroup condition | , is normal in | Step-fact direct | |

4 | and are conjugate in | is a WNSCDIN-subgroup of | Steps (2), (3) | Given-step direct | |

5 | is normal in | Step (4) | Given-step direct |