# Global 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

## Contents

## Definition

### Quick definition

A group is termed a **global Lazard Lie group** if its nilpotency class is finite and less than or equal to the group's powering threshold.

### Explicit definition

A group is termed a **global** class **Lazard Lie group** for some natural number if **both** the following hold:

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

1 | The powering threshold for is at least , i.e., is powered for the set of all primes less than or equal to . | is uniquely -divisible for all primes . In other words, if is a prime and , there is a unique value satisfying . |

2 | The nilpotency class of is at most . | 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.

A group is termed a global Lazard Lie group if it is a global class Lazard Lie group for some natural number .

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

### Set of possible values for which a group is a global class Lazard Lie group

A group is a global Lazard Lie group if and only if its nilpotency class is less than or equal to its powering threshold. The set of permissible values for which the group is a global class Lazard Lie group is the set of satisfying:

nilpotency class powering threshold

### p-group version

A p-group is termed a **global Lazard Lie group** if its nilpotency class is at most .

## Relation with other properties

### Stronger properties

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

rationally powered nilpotent group | |FULL LIST, MORE INFO | |||

p-group of nilpotency class less than p | |FULL LIST, MORE INFO | |||

Baer Lie group | class two global Lazard Lie group | |FULL LIST, MORE INFO | ||

class three Lazard Lie group | class three global Lazard Lie group | |FULL LIST, MORE INFO |

### Weaker properties

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

Lazard Lie group | |FULL LIST, MORE INFO | |||

nilpotent group | |FULL LIST, MORE INFO |