Global Lazard Lie group

From Groupprops
Jump to: navigation, search
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 properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

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

No. Shorthand for property Explanation
1 The powering threshold for G is at least c, i.e., G is powered for the set of all primes less than or equal to c. G is uniquely p-divisible for all primes p \le c. In other words, if p \le c is a prime and g \in G, there is a unique value h \in G satisfying h^p = g.
2 The nilpotency class of G is at most c. G is a nilpotent group of nilpotency class at most c.

Condition (1) gets more demanding (i.e., stronger, so satisfied by fewer groups) as c increases, while condition (2) gets less demanding (i.e., weaker, so satisfied by fewer groups) as we increase c. Thus, a particular value of c 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 c Lazard Lie group for some natural number c.

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 c for which a group is a global class c 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 c values for which the group is a global class c Lazard Lie group is the set of c satisfying:

nilpotency class \le c \le powering threshold

p-group version

A p-group is termed a global Lazard Lie group if its nilpotency class is at most p - 1.

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