# Lie ring of nilpotency class two

From Groupprops

ANALOGY: This is an analogue in Lie ring of a property encountered in group. Specifically, it is a Lie ring property analogous to the group property: group of nilpotency class two

View other analogues of group of nilpotency class two | View other analogues in Lie rings of group properties (OR, View as a tabulated list)

## Contents

## Definition

QUICK PHRASES: class two, inner derivation Lie ring is abelian, derived subring inside center, Lie brackets are central, triple Lie brackets are trivial

### Symbol-free definition

A group is said to be of **nilpotency class two** or **nilpotence class two** if it satisfies the following equivalent conditions:

- Its nilpotency class is at most two, i.e., it is nilpotent of class at most two.
- Its derived subring (i.e., the subring generated by all elements arising from the image of the Lie bracket mapping) is contained in its center
- The commutator of any two elements of the group is central
- Any triple commutator (i.e., a commutator where one of the terms is itself a commutator) gives the zero element
- Its Lie ring of inner derivations is abelian.

NOTE:nilpotency class twois occasionally used to refer to a Lie ring whose nilpotency class is precisely two, i.e., a non-abelian Lie ring whose nilpotency class is two. This is a more restrictive use of the term than the typical usage, which includes abelian Lie rings.

## Relation with other properties

### Stronger properties

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

abelian Lie ring | Lie bracket is trivial | LCS-Baer Lie ring, Lie ring arising as the double of a class two Lie cring, Lie ring arising as the skew of a class two near-Lie cring, Lie ring whose bracket is the double of a Lie bracket giving nilpotency class two|FULL LIST, MORE INFO | ||

Baer Lie ring | Lie ring of nilpotency class two where every element has a unique half | LCS-Baer Lie ring, LUCS-Baer Lie ring, Lie ring arising as the double of a class two Lie cring, Lie ring arising as the skew of a class two near-Lie cring, Lie ring whose bracket is the double of a Lie bracket giving nilpotency class two|FULL LIST, MORE INFO | ||

Lie ring whose bracket is the double of a Lie bracket giving nilpotency class two | Lie ring arising as the double of a class two Lie cring, Lie ring arising as the skew of a class two near-Lie cring|FULL LIST, MORE INFO |

### Weaker properties

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

nilpotent Lie ring | |FULL LIST, MORE INFO | |||

metabelian Lie ring | derived subring is abelian | 2-Engel Lie ring, Lie ring of nilpotency class three|FULL LIST, MORE INFO | ||

2-Engel Lie ring | Any triple commutator where two of the three inputs are equal must be trivial, i.e., for all . | |FULL LIST, MORE INFO |