UCS-Baer Lie group: Difference between revisions
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Finite UCS-Baer Lie groups coincide with finite Baer Lie groups, and therefore with [[odd-order class two group]]s. | Finite UCS-Baer Lie groups coincide with finite Baer Lie groups, and therefore with [[odd-order class two group]]s. | ||
==Examples== | |||
===Finite examples=== | |||
In the finite case, UCS-Baer Lie groups coincide with finite Baer Lie groups, and therefore with [[odd-order class two group]]s. | |||
===Infinite examples=== | |||
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 <math>\mathbb{Q} \otimes</math> the center. To be more precise, we do not need to use the whole field of rationals: taking the tensor product with <math>\mathbb{Z}[1/2]</math> should suffice. | |||
==Relation with other properties== | ==Relation with other properties== | ||
Revision as of 16:35, 2 July 2017
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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
Definition
A group is termed a UCS-Baer Lie group if is a group of nilpotency class two and the center is a 2-powered group.
UCS-Baer Lie groups can participate on the group side of the UCS-Baer correspondence; the objects on the Lie ring side are UCS-Baer Lie rings.
Finite UCS-Baer Lie groups coincide with finite Baer Lie groups, and therefore with odd-order class two groups.
Examples
Finite examples
In the finite case, UCS-Baer Lie groups coincide with finite Baer Lie groups, and therefore with odd-order class two groups.
Infinite examples
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 the center. To be more precise, we do not need to use the whole field of rationals: taking the tensor product with should suffice.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Baer Lie group | follows from center is local powering-invariant | central product of UT(3,Z) and Q | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| LUCS-Baer Lie group | every element in the derived subgroup has a unique square root in the center | obvious | central product of UT(3,Z) and Z identifying center with 2Z | |FULL LIST, MORE INFO |
| CS-Baer Lie group | intermediate subgroup between derived subgroup and center such that every element of derived subgroup has unique half in that subgroup | we can set the intermediate subgroup to be the whole center | central product of UT(3,Z) and Z identifying center with 2Z | |FULL LIST, MORE INFO |
| group of nilpotency class two | direct | any abelian group that is not 2-powered, such as cyclic group:Z2 or group of integers | |FULL LIST, MORE INFO |
Incomparable properties
| Property | Meaning | Proof that UCS-Baer Lie group may not have this property | Proof that a group with this property may not be a UCS-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 |