Baer Lie group: Difference between revisions

From Groupprops
No edit summary
 
(3 intermediate revisions by the same user not shown)
Line 6: Line 6:


# It is a [[defining ingredient::nilpotent group]] of [[defining ingredient::group of nilpotency class two|class two]], i.e., its [[defining ingredient::nilpotency class]] is at most two.
# It is a [[defining ingredient::nilpotent group]] of [[defining ingredient::group of nilpotency class two|class two]], i.e., its [[defining ingredient::nilpotency class]] is at most two.
# It is a [[defining ingredient::2-powered group]]: For every <math>g \in G</math>, there is a unique element <math>h \in G</math> such that <math>h^2 = g</math>.
# It is a [[defining ingredient::2-powered group]] (also known as uniquely 2-divisible group): For every <math>g \in G</math>, there is a unique element <math>h \in G</math> such that <math>h^2 = g</math>.


Given condition (1), condition (2) is equivalent to requiring that <math>G</math> be both [[2-torsion-free group|2-torsion-free]] (i.e., no element of order two) and [[2-divisible group|2-divisible]]. (see [[equivalence of definitions of nilpotent group that is torsion-free for a set of primes]]).
Given condition (1), condition (2) is equivalent to requiring that <math>G</math> be both [[2-torsion-free group|2-torsion-free]] (i.e., no element of order two) and [[2-divisible group|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]].
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]].
A finite group is a Baer Lie group if and only if it is an [[odd-order class two group]].
==Metaproperties==
{| class="sortable" border="1"
! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
|-
| [[dissatisfies metaproperty::subgroup-closed group property]] || No || [[Baer Lie property is not subgroup-closed]] || It is possible to have a Baer Lie group <math>G</math> and a subgroup <math>H</math> of <math>G</math> such that <math>H</math> is ''not'' a Baer Lie group in its own right.
|-
| [[dissatisfies metaproperty::quotient-closed group property]] || No || [[Baer Lie property is not quotient-closed]] || It is possible to have a Baer Lie group <math>G</math> and a [[normal subgroup]] <math>H</math> of <math>G</math> such that the [[quotient group]] <math>G/H</math> is ''not'' a Baer Lie group in its own right.
|-
| [[satisfies metaproperty::direct product-closed group property]] || Yes || [[Baer Lie property is direct product-closed]] || Given Baer Lie groups <math>G_i, i \in I</math> the [[external direct product]] of all the <math>G_i</math>s is also a Baer Lie group.
|}


==Relation with other properties==
==Relation with other properties==
Line 19: Line 33:
! 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
|-
|-
| [[Weaker than::odd-order abelian group]] || || || ||  
| [[Weaker than::odd-order abelian group]] || || || || {{intermediate notions short|Baer Lie group|odd-order abelian group}}
|-
|-
| [[Weaker than::odd-order class two group]] || group of odd order and nilpotency class two; equivalently, a ''finite'' Baer Lie group. || || ||
| [[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}}
|-
|-
| [[Weaker than::rationally powered class two group]] || || || ||
| [[Weaker than::rationally powered class two group]] || || || || {{intermediate notions short|Baer Lie group|rationally powered class two group}}
|}
|}


Line 31: Line 45:
! 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
|-
|-
| [[Stronger than::Lazard Lie group]] || || || ||
| [[Stronger than::Lazard Lie group]] || Powering threshold is greater than or equal to 3-local nilpotency class || || || {{intermediate notions short|Lazard Lie group|Baer Lie group}}
|-
| [[Stronger than::global Lazard Lie group]] || Powering threshold is greater than or equal to nilpotency class || || || {{intermediate notions short|global Lazard Lie group|Baer Lie group}}
|-
| [[Stronger than::UCS-Baer Lie group]] || class two, and center is 2-powered || || || {{intermediate notions short|UCS-Baer Lie group|Baer Lie group}}
|-
| [[Stronger than::LCS-Baer Lie group]] || class two, and derived subgroup is 2-powered || || || {{intermediate notions short|LCS-Baer Lie group|Baer Lie group}}
|-
| [[Stronger than::LUCS-Baer Lie group]] || class two, and every element in derived subgroup has unique square root in center || || || {{intermediate notions short|LUCS-Baer Lie group|Baer Lie group}}
|}
|}

Latest revision as of 16:10, 2 July 2017

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 (also known as uniquely 2-divisible group): For every g∈G, there is a unique element h∈G such that h2=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.

A finite group is a Baer Lie group if and only if it is an odd-order class two group.

Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
subgroup-closed group property No Baer Lie property is not subgroup-closed It is possible to have a Baer Lie group G and a subgroup H of G such that H is not a Baer Lie group in its own right.
quotient-closed group property No Baer Lie property is not quotient-closed It is possible to have a Baer Lie group G and a normal subgroup H of G such that the quotient group G/H is not a Baer Lie group in its own right.
direct product-closed group property Yes Baer Lie property is direct product-closed Given Baer Lie groups Gi,i∈I the external direct product of all the Gis is also a Baer Lie group.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
odd-order abelian 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 Powering threshold is greater than or equal to 3-local nilpotency class |FULL LIST, MORE INFO
global Lazard Lie group Powering threshold is greater than or equal to nilpotency class |FULL LIST, MORE INFO
UCS-Baer Lie group class two, and center is 2-powered |FULL LIST, MORE INFO
LCS-Baer Lie group class two, and derived subgroup is 2-powered |FULL LIST, MORE INFO
LUCS-Baer Lie group class two, and every element in derived subgroup has unique square root in center |FULL LIST, MORE INFO
Retrieved from "https://groupprops.subwiki.org/w/index.php?title=Baer_Lie_group&oldid=50466"