CS-Baer Lie group

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Definition

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

  1. G is a group of nilpotency class two, i.e., its nilpotency class is at most two. Equivalently, the derived subgroup [G,G] is contained in the center Z(G) of G.
  2. There exists a subgroup H of G such that [G,G] \le H \le Z(G), and every element of [G,G] has a unique square root within H.

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 G be the group central product of UT(3,Z) and Z identifying center with 2Z. Then, the group G \times \mathbb{Z}/2\mathbb{Z} 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

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two |FULL LIST, MORE INFO
group of nilpotency class two whose commutator map is the double of a skew-symmetric cyclicity-preserving 2-cocycle |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 Group of nilpotency class two whose commutator map is the double of a skew-symmetric cyclicity-preserving 2-cocycle, Group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two|FULL LIST, MORE INFO
group that is 1-isomorphic to an abelian group the group is 1-isomorphic to an abelian group Group of nilpotency class two whose commutator map is the double of a skew-symmetric cyclicity-preserving 2-cocycle, Group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two|FULL LIST, MORE INFO
group of nilpotency class two Group of nilpotency class two whose commutator map is the double of a skew-symmetric cyclicity-preserving 2-cocycle, Group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two, Group whose derived subgroup is contained in the square of its center|FULL LIST, MORE INFO