# Strongly paranormal implies paranormal

From Groupprops

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

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## Contents

## Statement

Any strongly paranormal subgroup of a group is paranormal.

## Definitions used

### Strongly paranormal subgroup

`Further information: strongly paranormal subgroup`

A subgroup is termed **strongly paranormal** if, for any , . Here, is the subgroup generated by all commutators between and elements of .

### Paranormal subgroup

`Further information: paranormal subgroup`

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

## Facts used

## Proof

### Proof idea

The key idea is to show, using the identity, that any normal subgroup of that contains must also contain , and hence must be the whole of .

### Proof details

**Given**: A group , a subgroup such that for any .

*To prove'*: If is a normal subgroup of that contains , .

**Proof**:

- : This follows from fact (1).
- : Since is in , we have . The latter is contained in , since is normal in .
- : This follows from the previous step, and the fact that .
- : This follows from the previous step, and the fact that .
- : We already have , and the previous fact yields . Thus, we get . By fact (1), this yields . Since , we get equality.