Difference between revisions of "Baer Lie group"

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(Relation with other properties)
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 
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| [[Weaker than::odd-order abelian group]] || || || ||  
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| [[Weaker than::odd-order abelian group]] || || || || {{intermediate notions short|Baer Lie group|odd-order abelian 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::odd-order class two group]] || group of odd order and nilpotency class two; equivalently, a ''finite'' Baer Lie group. || || || {{intermediate notions short|Baer Lie group|odd-order class two group}}
 
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| [[Weaker than::rationally powered class two group]] || || || ||
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| [[Weaker than::rationally powered class two group]] || || || || {{intermediate notions short|Baer Lie group|rationally powered class two group}}
 
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 
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| [[Stronger than::Lazard Lie group]] || || || ||
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| [[Stronger than::Lazard Lie group]] || || || || {{intermediate notions short|Lazard Lie group|Baer Lie group}}
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| [[Stronger than::global Lazard Lie group]] || || || || {{intermediate notions short|global Lazard Lie group|Baer Lie group}}
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| [[Stronger than::UCS-Baer Lie group]] || || || || {{intermediate notions short|UCS-Baer Lie group|Baer Lie group}}
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| [[Stronger than::LCS-Baer Lie group]] || || || || {{intermediate notions short|LCS-Baer Lie group|Baer Lie group}}
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| [[Stronger than::LUCS-Baer Lie group]] || || || || {{intermediate notions short|LUCS-Baer Lie group|Baer Lie group}}
 
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Revision as of 04:19, 6 August 2013

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 2-powered 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 2-torsion-free (i.e., no element of order two) and 2-divisible. (see equivalence of definitions of nilpotent group that is torsion-free for a set of primes).

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|FULL LIST, MORE INFO
odd-order class two group group of odd order and nilpotency class two; equivalently, a finite Baer Lie group. |FULL LIST, MORE INFO
rationally powered class two group |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Lazard Lie group Global Lazard Lie group|FULL LIST, MORE INFO
global Lazard Lie group |FULL LIST, MORE INFO
UCS-Baer Lie group |FULL LIST, MORE INFO
LCS-Baer Lie group |FULL LIST, MORE INFO
LUCS-Baer Lie group UCS-Baer Lie group|FULL LIST, MORE INFO