LCS-Baer Lie group
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A LCS-Baer Lie group or lower central series Baer Lie group is a group satisfying both the following properties:
- It is a group of nilpotency class two, i.e., its nilpotency class is at most two.
- Its derived subgroup is a 2-powered group, i.e., a uniquely 2-divisible group. Note that since the group has class at most two, the derived subgroup must also be abelian.
Definition in terms of LCS-Lazard Lie group
A finite group is a LCS-Baer Lie group if and only if it is a group of nilpotency class (at most) two and its 2-Sylow subgroup is abelian.
The finite LCS-Baer Lie groups are the groups of nilpotency class two whose 2-Sylow subgroup is abelian. In particular, when the 2-Sylow subgroup is nontrivial abelian, these examples are not Baer Lie groups.
Any infinite Baer Lie group gives an example. In addition, examples like direct product of UT(3,Q) and Z are examples of LCS-Baer Lie groups that are not Baer Lie groups. The reason it fails to be a Baer Lie group is that there is a separate part of the center (outside the derived subgroup) that is not 2-divisible.
Relation with other properties
|Property||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|
|Baer Lie group||class at most two, and whole group is uniquely 2-divisible||cyclic group:Z2, or direct product of UT(3,Q) and Z|||FULL LIST, MORE INFO|
|abelian group||class at most one|||FULL LIST, MORE INFO|
|Property||Meaning||Proof that LCS-Baer Lie group may not have this property||Proof that a group with this property may not be a LCS-Baer Lie group|
|UCS-Baer Lie group||center is 2-powered||any abelian group with 2-torsion, such as cyclic group:Z2||central product of UT(3,Z) and Q|
|LUCS-Baer Lie group||derived subgroup has unique square roots in center||any abelian group with 2-torsion, such as cyclic group:Z2||central product of UT(3,Z) and Q|