UCS-Baer Lie group

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Definition

A group $G$ is termed a UCS-Baer Lie group if $G$ is a group of nilpotency class two and the center $Z(G)$ 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 $\mathbb{Q} \otimes$ 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.

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

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
LUCS-Baer Lie group every element in the derived subgroup has a unique square root in the center obvious central product of UT(3,Z) and Z identifying center with 2Z |FULL LIST, MORE INFO
CS-Baer Lie group intermediate subgroup between derived subgroup and center such that every element of derived subgroup has unique half in that subgroup we can set the intermediate subgroup to be the whole center central product of UT(3,Z) and Z identifying center with 2Z LUCS-Baer Lie group|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, LUCS-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, LUCS-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, LUCS-Baer Lie group|FULL LIST, MORE INFO
group 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, LUCS-Baer Lie group|FULL LIST, MORE INFO
group of nilpotency class two direct any abelian group that is not 2-powered, such as cyclic group:Z2 or group of integers 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, LUCS-Baer Lie group|FULL LIST, MORE INFO

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