# LCS-Baer Lie group

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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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

### 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 CS-Baer Lie group|FULL LIST, MORE INFO
group of nilpotency class two whose commutator map is the double of a skew-symmetric cyclicity-preserving 2-cocycle CS-Baer Lie group|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 CS-Baer Lie group, Group of nilpotency class two whose commutator map is the double of a skew-symmetric cyclicity-preserving 2-cocycle, Group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two|FULL LIST, MORE INFO
group 1-isomorphic to an abelian group the group is 1-isomorphic to an abelian group CS-Baer Lie group, Group of nilpotency class two whose commutator map is the double of a skew-symmetric cyclicity-preserving 2-cocycle, Group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two, Group of nilpotency class two whose commutator map is the skew of a cyclicity-preserving 2-cocycle|FULL LIST, MORE INFO
group of nilpotency class two CS-Baer Lie group, Group of nilpotency class two whose commutator map is the double of a skew-symmetric cyclicity-preserving 2-cocycle, Group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two, Group of nilpotency class two whose commutator map is the skew of a cyclicity-preserving 2-cocycle, Group whose derived subgroup is contained in the square of its center|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