# Paranormal implies polynormal

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., paranormal subgroup) must also satisfy the second subgroup property (i.e., polynormal subgroup)

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## Contents

## Statement

Any paranormal subgroup of a group is also a polynormal subgroup.

## Definitions used

For these definitions, denotes the conjugate of by , using the right-action convention (the action convention doesn't really matter). For subgroups , is the smallest subgroup of containing and invariant under the action of by conjugation.

### Paranormal subgroup

`Further information: Paranormal subgroup`

A subgroup of a group is termed **paranormal** in if, for any , is a contranormal subgroup of .

### Polynormal subgroup

`Further information: Polynormal subgroup`

A subgroup of a group is termed **polynormal** in if, for any , is a contranormal subgroup of .

## Facts used

## Proof

**Given**: A paranormal subgroup of a group .

**To prove**: For any , is contranormal in .

**Proof**: Clearly, is generated by for all , which in turn means that it is generated by the subgroups . is contranormal in each of these by definition of paranormality, so by fact (1), is contranormal in .