# P-group not implies nilpotent

This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., p-group) neednotsatisfy the second group property (i.e., nilpotent group)

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## Contents

## Statement

A p-group (i.e., a possibly infinite group in which the order of every element is the power of a fixed prime ) need not be nilpotent.

## Related facts

### Similar facts

- p-group not implies solvable: This is true, at least for large enough primes, where we can take the Tarski groups.

### Opposite facts

- Prime power order implies nilpotent: Any
*finite*-group (which is the same as a group of prime power order) must be nilpotent.`For full proof, refer: Prime power order implies nilpotent` - Locally finite Artinian p-group implies hypercentral

## Proof

### McLain's example

`Further information: restricted regular wreath product of group of prime order and quasicyclic group`

For the given prime , let be the quasicyclic group for ; concretely, is the group of roots of unity in for all nonnegative integers . Clearly, is a -group.

Let be the restricted regular wreath product of the group of prime order with . In other words, is the restricted external wreath product of the group of prime order with having the left regular action. Equivalently, is the semidirect product of the additive group of the group ring by acting via left multiplication. We claim the following:

- is a -group: is a semidirect product of two -groups. In particular, it is the extension of one -group (the additive group of the group ring) by another (the multiplicative group living as a subgroup of the group of units of the group ring); hence it is a -group.
- is a metabelian group: The derived length of is two. In fact, the additive group is an abelian normal subgroup of with Abelian quotient.
- is centerless: This is clear by inspection.

### Tarski's examples

For any prime for which a Tarski group exists, the Tarski group is an example of a p-group that is not nilpotent. In fact, it is not even solvable.

Tarski groups do not exist for all primes.