# LUCS-Baer Lie group

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## 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:

1. Every element of its derived subgroup has a unique square root in the whole group.
2. Every element of its derived subgroup has a unique square root among the elements in its center.
3. 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:

## 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 $\mathbb{Q} \otimes$ the center. To be more precise, we do not need to use the whole field of rationals: taking the tensor product with $\mathbb{Z}[1/2]$ should suffice.

#### Examples that are not UCS

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

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
CS-Baer Lie group any abelian group with 2-torsion, such as cyclic group:Z2 |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|FULL LIST, MORE INFO
group that is 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|FULL LIST, MORE INFO
group of nilpotency class two (via CS-Baer Lie 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 whose derived subgroup is contained in the square of its center|FULL LIST, MORE INFO