# 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

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Definition

### Direct definition

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

- It is a group of nilpotency class two, i.e., its nilpotency class is at most two.
- 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

### 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 |