UCS-Baer Lie group - Revision history

Vipul: /* Weaker properties */

2017-07-02T17:12:42Z

Weaker properties

← Older revision		Revision as of 17:12, 2 July 2017
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	\|-		\|-
	\| [[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::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}}		\| [[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}}

Vipul at 16:35, 2 July 2017

2017-07-02T16:35:15Z

← Older revision		Revision as of 16:35, 2 July 2017
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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==

Vipul: /* Incomparable properties */

2017-07-02T16:25:45Z

Incomparable properties

← Older revision		Revision as of 16:25, 2 July 2017
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	===Incomparable properties===		===Incomparable properties===


	{\| class="sortable" border="1"		{\| 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		! 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]]
	\|}		\|}

Vipul at 16:19, 2 July 2017

2017-07-02T16:19:06Z

← Older revision		Revision as of 16:19, 2 July 2017
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	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]].		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.

	==Relation with other properties==		==Relation with other properties==

Vipul at 16:17, 2 July 2017

2017-07-02T16:17:34Z

← Older revision		Revision as of 16:17, 2 July 2017
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	! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	\|-		\|-
	\| [[Weaker than::Baer Lie group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|UCS-Baer Lie group\|Baer Lie group}}		\| [[Weaker than::Baer Lie group]] \|\| \|\| follows from [[center is local powering-invariant]] \|\| [[central product of UT(3,Z) and Q]] \|\| {{intermediate notions short\|UCS-Baer Lie group\|Baer Lie group}}
	\|}		\|}

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	! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	\|-		\|-
	\| [[Stronger than::LUCS-Baer Lie group]] \|\| every element in the derived subgroup has a unique square root in the center \|\| \|\| \|\| {{intermediate notions short\|LUCS-Baer Lie group\|UCS-Baer Lie group}}		\| [[Stronger than::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]] \|\| {{intermediate notions short\|LUCS-Baer Lie group\|UCS-Baer Lie group}}
	\|-		\|-
	\| [[Stronger than::CS-Baer Lie group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|CS-Baer Lie group\|UCS-Baer Lie group}}		\| [[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]] \|\| \|\| \|\| \|\| {{intermediate notions short\|group of nilpotency class two\|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}}
			\|}

			===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
	\|}		\|}

Vipul: Created page with "{{wikilocal}} {{group property}} ==Definition== A group $G$ is termed a '''UCS-Baer Lie group''' if $G$ is a group of nilpotency class two and ..."

2013-07-11T01:00:13Z

Created page with "{{wikilocal}} {{group property}} ==Definition== A group <math>G</math> is termed a '''UCS-Baer Lie group''' if <math>G</math> is a group of nilpotency class two and ..."

New page

{{wikilocal}}
{{group property}}

==Definition==

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]].

==Relation with other properties==

===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::Baer Lie group]] || || || || {{intermediate notions short|UCS-Baer Lie group|Baer Lie group}}
|}

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::LUCS-Baer Lie group]] || every element in the derived subgroup has a unique square root in the center || || || {{intermediate notions short|LUCS-Baer Lie group|UCS-Baer Lie group}}
|-
| [[Stronger than::CS-Baer Lie group]] || || || || {{intermediate notions short|CS-Baer Lie group|UCS-Baer Lie group}}
|-
| [[Stronger than::group of nilpotency class two]] || || || || {{intermediate notions short|group of nilpotency class two|UCS-Baer Lie group}}
|}