# Paranormal implies weakly closed in intermediate nilpotent

From Groupprops

## Contents

## Statement

### Statement with symbols

Suppose are groups such that:

- is a Paranormal subgroup (?) of .
- is a Nilpotent group (?).

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

## Related facts

### Corollaries

## 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: Paranormal subgroup`

A subgroup of a group is termed **paranormal** in if for any , is a contranormal subgroup of ; in other words, the normal closure of in is the whole of .

### 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

## Proof

The proof follows directly from facts (1) and (2).