# Weakly normal implies weakly closed in intermediate nilpotent

## Contents

## Statement

### Statement with symbols

Suppose are groups such that:

- is a Weakly normal subgroup (?) of .
- is a Nilpotent group (?).

Then, is a Weakly closed subgroup (?) of .

## Related facts

### Corollaries

- Paranormal implies weakly closed in intermediate nilpotent
- Pronormal implies weakly closed in intermediate nilpotent

## Definitions used

For these definitions, denotes the conjugate subgroup by . (This is the right-action convention; however, adopting a left-action convention does not alter any of the proof details).

### Paranormal subgroup

`Further information: Weakly normal subgroup`

A subgroup of a group is termed **paranormal** in if for any , implies . In other words, it is weakly closed in its normalizer.

### Weakly closed subgroup

`Further information: Weakly closed subgroup`

Suppose are groups. We say is weakly closed in with respect to if, for any such that , we have .

## Facts used

- Weakly normal implies intermediately subnormal-to-normal
- Nilpotent implies every subgroup is subnormal

## Proof

**Given**: with a paranormal subgroup of and a nilpotent group.

**To prove**: For any such that , we have .

**Proof**: By fact (2), is a subnormal subgroup of . By fact (1), is therefore normal in . Thus, .

Since is weakly closed in by definition, whenever , we get . In particular, whenever , we get , so is weakly closed in .