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:

• (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 $\mathbb{Q} \otimes$ 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

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

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
2-torsion-free group				|FULL LIST, MORE INFO

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