# UCS-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

A group is termed a **UCS-Baer Lie group** if is a group of nilpotency class two and the center is a 2-powered group.

UCS-Baer Lie groups can participate on the group side of the UCS-Baer correspondence; the objects on the Lie ring side are UCS-Baer Lie rings.

Finite UCS-Baer Lie groups coincide with finite Baer Lie groups, and therefore with odd-order class two groups.

## Examples

### Finite examples

In the finite case, UCS-Baer Lie groups coincide with finite Baer Lie groups, and therefore with odd-order class two groups.

### Infinite examples

An example is central product of UT(3,Z) and Q.

More generally, for a finitely generated torsion-free nilpotent group of class two, we can construct an example by taking the central product of that group with the vector space over the rationals obtained by taking the center. To be more precise, we do not need to use the whole field of rationals: taking the tensor product with should suffice.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Baer Lie group | follows from center is local powering-invariant | central product of UT(3,Z) and Q | |FULL LIST, MORE INFO |

### Weaker properties

### Incomparable properties

Property | Meaning | Proof that UCS-Baer Lie group may not have this property | Proof that a group with this property may not be a UCS-Baer Lie group |
---|---|---|---|

LCS-Baer Lie group | derived subgroup is 2-powered | central product of UT(3,Z) and Q | any abelian group with 2-torsion, such as cyclic group:Z2 |