Difference between revisions of "Baer Lie group"
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# 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::uniquely 2-divisible group | + | # 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 [[torsion-free group|torsion-free]] (i.e., | + | 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== | ||
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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 | ||
|- | |- | ||
− | | [[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}} |
|} | |} | ||
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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 | ||
|- | |- | ||
− | | [[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 satisfying the following two conditions:
- It is a nilpotent group of class two, i.e., its nilpotency class is at most two.
- It is a 2-powered group (also known as uniquely 2-divisible group): For every , there is a unique element such that .
Given condition (1), condition (2) is equivalent to requiring that 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 and a subgroup of such that 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 and a normal subgroup of such that the quotient group 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 the external direct product of all the s 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 | 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 | Powering threshold is greater than or equal to 3-local nilpotency class | Global Lazard Lie group|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 | UCS-Baer Lie group|FULL LIST, MORE INFO |