# LUCS-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 **LUCS-Baer Lie group** if it is a group of nilpotency class two (i.e., its derived subgroup is contained in ts center) and it satisfies the following equivalent conditions:

- Every element of its derived subgroup has a unique square root in the whole group.
- Every element of its derived subgroup has a unique square root among the elements in its center.
- Every element of the derived subgroup has a unique square root in the whole group
*and*that square root is in the center.

### Equivalence of definitions

`Further information: equivalence of definitions of LUCS-Baer Lie group`

Note that (3) implies both (1) and (2). The reverse implications are somewhat harder:

- (1) implies (3): This follows from the fact that center is local powering-invariant. Note that we use class two in order to observe that the element whose square root we are taking is in the center to begin with.
- (2) implies (3): This follows from the equivalence of definitions of nilpotent group that is torsion-free for a set of primes.

## Examples

### Finite examples

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

### Infinite examples

#### Examples that are UCS

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.

#### Examples that are not UCS

An example is central product of UT(3,Z) and Z identifying center with 2Z.

Conceptually, instead of taking the central product with a free module over the full ring (so that we can divide by arbitrarily large powers of 2) we can simply take a central product that allows for one additional stage of division.

## Relation with other properties

### Stronger properties

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

Baer Lie group | 2-powered group of nilpotency class two | central product of UT(3,Z) and Z identifying center with 2Z | UCS-Baer Lie group|FULL LIST, MORE INFO | |

UCS-Baer Lie group | class at most two and the center is 2-powered. | central product of UT(3,Z) and Z identifying center with 2Z | |FULL LIST, MORE INFO |

### Weaker properties

### Incomparable properties

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