CS-Baer Lie group
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A CS-Baer Lie group is a group satisfying both the following two conditions:
- is a group of nilpotency class two, i.e., its nilpotency class is at most two. Equivalently, the derived subgroup is contained in the center of .
- There exists a subgroup of such that , and every element of has a unique square root within .
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 be the group central product of UT(3,Z) and Z identifying center with 2Z. Then, the group 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
|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|