# Inner-Lazard Lie group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## Definition

### Quick definition

A group is termed a **Lazard Lie group** if its 3-local nilpotency class is finite and less than or equal to (the group's powering threshold + 1).

### Explicit definition

A group is termed an **inner-Lazard Lie group** if there is a natural number such that **both** the following hold:

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

1 | is powered for the set of all primes strictly less than . |
is uniquely -divisible for all primes . In other words, if is a prime and , there is a unique value satisfying . |

2 | The 3-local nilpotency class of is at most . | For any three elements of , the subgroup of generated by these three elements is a nilpotent group 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 may work for a group but larger and smaller values may not.

### p-group version

A p-group is termed an **inner-Lazard Lie group** if its 3-local nilpotency class is at most . In other words, every subgroup of it generated by at most three elements has nilpotency class at most where is the prime associated with the group.