UCS-Baer Lie group: Difference between revisions

Latest revision as of 17:12, 2 July 2017

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Definition

A group $G$ is termed a UCS-Baer Lie group if $G$ is a group of nilpotency class two and the center $Z (G)$ 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 $Q \otimes$ the center. To be more precise, we do not need to use the whole field of rationals: taking the tensor product with $Z [1 / 2]$ 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 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 1-isomorphic to an abelian group	the group is 1-isomorphic to an abelian group			\|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

@@ Line 5: / Line 5: @@
 A [[group]] <math>G</math> is termed a '''UCS-Baer Lie group''' if <math>G</math> is a [[group of nilpotency class two]] and the [[center]] <math>Z(G)</math> is a [[powered group for a set of primes|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 ring]]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==
@@ Line 24: / Line 40: @@
 |-
 | [[Stronger than::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]] || {{intermediate notions short|CS-Baer Lie group|UCS-Baer Lie group}}
+|-
+| [[Stronger than::group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two]] || || || || {{intermediate notions short|group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two|UCS-Baer Lie group}}
+|-
+| [[Stronger than::group of nilpotency class two whose commutator map is the double of a skew-symmetric cyclicity-preserving 2-cocycle]] || || || || {{intermediate notions short|group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two|UCS-Baer Lie group}}
+|-
+| [[Stronger than::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 || || || {{intermediate notions short|group whose derived subgroup is contained in the square of its center|UCS-Baer Lie group}}
+|-
+| [[Stronger than::group 1-isomorphic to an abelian group]] || the group is [[1-isomorphic groups|1-isomorphic]] to an abelian group || || || {{intermediate notions short|group 1-isomorphic to an abelian group|UCS-Baer Lie group}}
 |-
 | [[Stronger than::group of nilpotency class two]] || || direct || any abelian group that is not 2-powered, such as [[cyclic group:Z2]] or [[group of integers]] || {{intermediate notions short|group of nilpotency class two|UCS-Baer Lie group}}
@@ Line 29: / Line 53: @@
 ===Incomparable properties===
 {| class="sortable" border="1"
 ! 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
+| [[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]]
 |}