Difference between revisions of "Baer Lie group"

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(Relation with other properties)
(Relation with other properties)
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| [[Weaker than::odd-order abelian group]] || || || ||  
 
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| [[Weaker than::odd-order class two group]] || || || ||
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| [[Weaker than::odd-order class two group]] || group of odd order and nilpotency class two; equivalently, a ''finite'' Baer Lie group. || || ||
 
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| [[Weaker than::rationally powered class two group]] || || || ||
 
| [[Weaker than::rationally powered class two group]] || || || ||

Revision as of 20:39, 10 February 2011

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: group of nilpotency class and uniquely 2-divisible group
View other group property conjunctions OR view all group properties

Definition

A Baer Lie group is a group G satisfying the following two conditions:

  1. It is a nilpotent group of class two, i.e., its nilpotency class is at most two.
  2. It is a uniquely 2-divisible group: For every g \in G, there is a unique element h \in G such that h^2 = g.

Given condition (1), condition (2) is equivalent to requiring that G be both torsion-free (i.e., aperiodic) and 2-divisible.

A Baer Lie group is a group that can serve on the group side of a Baer correspondence, i.e., it has a corresponding Baer Lie ring.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
odd-order abelian group
odd-order class two group group of odd order and nilpotency class two; equivalently, a finite Baer Lie group.
rationally powered class two group

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Lazard Lie group