# Lazard Lie ring

## Contents

## Definition

A **Lazard Lie ring** is a Lie ring such that there exists a natural number with **both** the following two properties:

No. | Shorthand for property | Explanation |
---|---|---|

1 | The additive group is a powered group for the set of all primes less than or equal to . | For any prime number , and any element , there is a unique element such that . |

2 | The 3-local nilpotency class is at most . | For any subset of of size at most three, the subring of generated by that subset is a nilpotent Lie ring of nilpotency class at most . |

Condition (1) gets more demanding (i.e., stronger, so satisfied by fewer groups) as increases, while condition (2) gets less demanding (i.e., weaker, so satisfied by fewer groups) as we increase . Thus, a particular value of c may work for a Lie ring but larger and smaller values may not.

A Lazard Lie ring is a Lie ring that can participate as the Lie ring side of a Lazard correspondence. The other side of the correspondence is its Lazard Lie group.

### p-Lie ring

A **Lazard -Lie ring** is a special case of the above, namely a Lie ring such that:

- There is a prime such that every element of has order a power of .
- The Lie subring of generated by any three elements of is a nilpotent Lie ring of nilpotency class less than .

A Lazard Lie ring is a -Lie ring that can participate as the Lie ring side of a Lazard correspondence. The other side of the correspondence is its Lazard Lie group, which is a p-group.