# CS-Baer Lie group

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

## Contents

## Definition

A **CS-Baer Lie group** is a group satisfying **both** the following two conditions:

- is a group of nilpotency class two, i.e., its nilpotency class is at most two. Equivalently, the derived subgroup is contained in the center of .
- There exists a subgroup of such that , and every element of has a unique square root within .

## Examples

### Finite examples

For finite groups, being a CS-Baer Lie group is equivalent to being a LCS-Baer Lie group, which in turn means that it is a group of class two whose 2-Sylow subgroup is abelian.

### Infinite examples

There are examples of infinite groups that are CS-Baer but not LCS-Baer. The smallest example is central product of UT(3,Z) and Z identifying center with 2Z. Note that this particular example is a LUCS-Baer Lie group (see LUCS-Baer Lie group#Examples).

We can create examples of CS-Baer Lie groups that are neither LUCS-Baer Lie groups nor LCS-Baer Lie groups, by combining features of the above examples. Specifically, let be the group central product of UT(3,Z) and Z identifying center with 2Z. Then, the group is a CS-Baer Lie group but it fails to be a LUCS-Baer Lie group (since it has 2-torsion within the center). It also fails to be a LCS-Baer Lie group (since the derived subgroup itself does not allow for halving).

## Relation with other properties

### Stronger properties

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

abelian group | LCS-Baer Lie group|FULL LIST, MORE INFO | |||

Baer Lie group | 2-powered and class two | cyclic group:Z2 (or any abelian group with 2-torsion) | LCS-Baer Lie group, LUCS-Baer Lie group, UCS-Baer Lie group|FULL LIST, MORE INFO | |

LCS-Baer Lie group | class two and derived subgroup is 2-powered | central product of UT(3,Z) and Q | |FULL LIST, MORE INFO | |

UCS-Baer Lie group | class two and center is 2-powered | cyclic group:Z2 (or any abelian group with 2-torsion) | LUCS-Baer Lie group|FULL LIST, MORE INFO | |

LUCS-Baer Lie group | class two and derived subgroup elements have unique square roots in center | cyclic group:Z2 (or any abelian group with 2-torsion) | |FULL LIST, MORE INFO |