CS-Baer Lie group
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Definition
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 .
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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 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
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| abelian group | |FULL LIST, MORE INFO | |||
| Baer Lie group | 2-powered and class two | cyclic group:Z2 (or any abelian group with 2-torsion) | |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) | |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 | |FULL LIST, MORE INFO | ||
| group that is 1-isomorphic to an abelian group | the group is 1-isomorphic to an abelian group | |FULL LIST, MORE INFO | ||
| group of nilpotency class two | |FULL LIST, MORE INFO |