# 2-locally nilpotent group

From Groupprops

## Contents

## Definition

A group is termed a **2-locally nilpotent group** if every 2-generated subgroup of it is a nilpotent group.

The 2-local nilpotency class of a 2-locally nilpotent group is defined as the supremum, over all 2-generated subgroups, of their nilpotency class. The 2-local nilpotency class of a 2-locally nilpotent group may be infinite. An example is the generalized dihedral group for 2-quasicyclic group.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

nilpotent group | 3-locally nilpotent group, Locally nilpotent group|FULL LIST, MORE INFO | |||

locally nilpotent group | every finitely generated subgroup is nilpotent | 3-locally nilpotent group|FULL LIST, MORE INFO | ||

3-locally nilpotent group | every subgroup generated by three elements is nilpotent | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Engel group | |FULL LIST, MORE INFO |