# Locally nilpotent 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., locally nilpotent group) neednotsatisfy the second group property (i.e., nilpotent group)

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## Contents

## Statement

It is possible to have a group that is locally nilpotent, i.e., every finitely generated subgroup of it is nilpotent, but itself is not nilpotent.

## Proof

### Example that satisfies the normalizer condition

The generalized dihedral group for 2-quasicyclic group is an example. This is obtained by taking the 2-quasicyclic group (the group of all roots of unity for arbitrary nonnegative integers ) and then constructing the generalized dihedral group for it: the semidirect product with the automorphism that sends each element to its negative.

This is a group satisfying normalizer condition.

### Example that does not satisfy the normalizer condition

`Further information: locally nilpotent not implies normalizer condition`

McLain's group over a field (a unitriangular matrix group with indexing set the rationals) gives an example of a locally nilpotent group that does not satisfy the normalizer condition.