LCS-Baer Lie group: Difference between revisions

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# It is a [[defining ingredient::group of nilpotency class two]], i.e., its [[nilpotency class]] is at most two.
# It is a [[defining ingredient::group of nilpotency class two]], i.e., its [[nilpotency class]] is at most two.
# Its [[derived subgroup]] is a uniquely 2-divisible group. Note that since the group has class at most two, the derived subgroup must also be abelian.
# Its [[derived subgroup]] is a [[2-powered group]], i.e., a uniquely 2-divisible group. Note that since the group has class at most two, the derived subgroup must also be abelian.


===Definition in terms of LCS-Lazard Lie group===
===Definition in terms of LCS-Lazard Lie group===
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A LCS-Baer Lie group can serve on the ''group'' side of the [[LCS-Baer correspondence]] (the other side is the [[LCS-Baer Lie ring]]).
A LCS-Baer Lie group can serve on the ''group'' side of the [[LCS-Baer correspondence]] (the other side is the [[LCS-Baer Lie ring]]).
A finite group is a LCS-Baer Lie group if and only if it is a group of nilpotency class (at most) two and its 2-Sylow subgroup is abelian.
== Examples ==
=== Finite examples ===
The finite LCS-Baer Lie groups are the groups of nilpotency class two whose 2-Sylow subgroup is abelian. In particular, when the 2-Sylow subgroup is nontrivial abelian, these examples are ''not'' Baer Lie groups.
=== Infinite examples ===
Any infinite Baer Lie group gives an example. In addition, examples like [[direct product of UT(3,Q) and Z]] are examples of LCS-Baer Lie groups that are not Baer Lie groups. The reason it fails to be a Baer Lie group is that there is a separate part of the center (outside the derived subgroup) that is not 2-divisible.


==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::Baer Lie group]] || class at most two, and ''whole'' group is uniquely 2-divisible || || || {{intermediate notions short|LCS-Baer Lie group|Baer Lie group}}
| [[Weaker than::Baer Lie group]] || class at most two, and ''whole'' group is uniquely 2-divisible || || [[cyclic group:Z2]], or [[direct product of UT(3,Q) and Z]] || {{intermediate notions short|LCS-Baer Lie group|Baer Lie group}}
|-
|-
| [[Weaker than::abelian group]] || class at most one || || || {{intermediate notions short|LCS-Baer Lie group|abelian group}}
| [[Weaker than::abelian group]] || class at most one || || || {{intermediate notions short|LCS-Baer Lie group|abelian 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::CS-Baer Lie group]] || class at most two, and some intermediate subgroup between derived subgroup and center where every element of derived subgroup has a unique half || || || {{intermediate notions short|CS-Baer Lie group|LCS-Baer Lie group}}
| [[Stronger than::CS-Baer Lie group]] || class at most two, and some intermediate subgroup between derived subgroup and center where every element of derived subgroup has a unique half || || [[central product of UT(3,Z) and Q]] (an example that is in fact a [[UCS-Baer Lie group]]) || {{intermediate notions short|CS-Baer Lie group|LCS-Baer Lie group}}
|-
|-
| [[Stronger than::group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two]] || || || || {{intermediate notions short|group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two|LCS-Baer Lie group}}
| [[Stronger than::group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two]] || || || || {{intermediate notions short|group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two|LCS-Baer Lie group}}
|-
| [[Stronger than::group of nilpotency class two whose commutator map is the double of a skew-symmetric cyclicity-preserving 2-cocycle]] || || || || {{intermediate notions short|group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two|LCS-Baer Lie group}}
|-
| [[Stronger than::group whose derived subgroup is contained in the square of its center]] || every element of the derived subgroup has a square root in the center || || || {{intermediate notions short|group whose derived subgroup is contained in the square of its center|LCS-Baer Lie group}}
|-
| [[Stronger than::group 1-isomorphic to an abelian group]] || the group is [[1-isomorphic groups|1-isomorphic]] to an abelian group || || || {{intermediate notions short|group 1-isomorphic to an abelian group|LCS-Baer Lie group}}
|-
|-
| [[Stronger than::group of nilpotency class two]] || || || || {{intermediate notions short|group of nilpotency class two|LCS-Baer Lie group}}
| [[Stronger than::group of nilpotency class two]] || || || || {{intermediate notions short|group of nilpotency class two|LCS-Baer Lie group}}
|}
=== Incomparable properties ===
{| class="sortable" border="1"
! Property !! Meaning !! Proof that LCS-Baer Lie group may not have this property !! Proof that a group with this property may not be a LCS-Baer Lie group
|-
| [[UCS-Baer Lie group]] || center is 2-powered || any [[abelian group]] with 2-torsion, such as [[cyclic group:Z2]] || [[central product of UT(3,Z) and Q]]
|-
| [[LUCS-Baer Lie group]] || derived subgroup has unique square roots in center || any [[abelian group]] with 2-torsion, such as [[cyclic group:Z2]] || [[central product of UT(3,Z) and Q]]
|}
|}

Latest revision as of 17:10, 2 July 2017

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Definition

Direct definition

A LCS-Baer Lie group or lower central series Baer Lie group is a group satisfying both the following properties:

  1. It is a group of nilpotency class two, i.e., its nilpotency class is at most two.
  2. Its derived subgroup is a 2-powered group, i.e., a uniquely 2-divisible group. Note that since the group has class at most two, the derived subgroup must also be abelian.

Definition in terms of LCS-Lazard Lie group

A LCS-Baer Lie group is a LCS-Lazard Lie group that is also a group of nilpotency class two.

A LCS-Baer Lie group can serve on the group side of the LCS-Baer correspondence (the other side is the LCS-Baer Lie ring).

A finite group is a LCS-Baer Lie group if and only if it is a group of nilpotency class (at most) two and its 2-Sylow subgroup is abelian.

Examples

Finite examples

The finite LCS-Baer Lie groups are the groups of nilpotency class two whose 2-Sylow subgroup is abelian. In particular, when the 2-Sylow subgroup is nontrivial abelian, these examples are not Baer Lie groups.

Infinite examples

Any infinite Baer Lie group gives an example. In addition, examples like direct product of UT(3,Q) and Z are examples of LCS-Baer Lie groups that are not Baer Lie groups. The reason it fails to be a Baer Lie group is that there is a separate part of the center (outside the derived subgroup) that is not 2-divisible.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Baer Lie group class at most two, and whole group is uniquely 2-divisible cyclic group:Z2, or direct product of UT(3,Q) and Z |FULL LIST, MORE INFO
abelian group class at most one |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
CS-Baer Lie group class at most two, and some intermediate subgroup between derived subgroup and center where every element of derived subgroup has a unique half central product of UT(3,Z) and Q (an example that is in fact a UCS-Baer Lie group) |FULL LIST, MORE INFO
group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two |FULL LIST, MORE INFO
group of nilpotency class two whose commutator map is the double of a skew-symmetric cyclicity-preserving 2-cocycle |FULL LIST, MORE INFO
group whose derived subgroup is contained in the square of its center every element of the derived subgroup has a square root in the center |FULL LIST, MORE INFO
group 1-isomorphic to an abelian group the group is 1-isomorphic to an abelian group |FULL LIST, MORE INFO
group of nilpotency class two |FULL LIST, MORE INFO

Incomparable properties

Property Meaning Proof that LCS-Baer Lie group may not have this property Proof that a group with this property may not be a LCS-Baer Lie group
UCS-Baer Lie group center is 2-powered any abelian group with 2-torsion, such as cyclic group:Z2 central product of UT(3,Z) and Q
LUCS-Baer Lie group derived subgroup has unique square roots in center any abelian group with 2-torsion, such as cyclic group:Z2 central product of UT(3,Z) and Q
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