Lazard Lie group - Revision history

Vipul: /* Explicit definition */

2017-07-02T00:18:31Z

Explicit definition

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	\|}		\|}

	Condition (1) gets more demanding (i.e., stronger, so satisfied by fewer groups) as <math>c</math> increases, while condition (2) gets less demanding (i.e., weaker, so satisfied by ~~fewer~~ groups) as we increase <math>c</math>. Thus, a particular value of <math>c</math> may work for a group but larger and smaller values may not.		Condition (1) gets more demanding (i.e., stronger, so satisfied by fewer groups) as <math>c</math> increases, while condition (2) gets less demanding (i.e., weaker, so satisfied by more groups) as we increase <math>c</math>. Thus, a particular value of <math>c</math> may work for a group but larger and smaller values may not.

	A group is termed a Lazard Lie group if it is a class <math>c</math> Lazard Lie group for some natural number <math>c</math>.		A group is termed a Lazard Lie group if it is a class <math>c</math> Lazard Lie group for some natural number <math>c</math>.

Vipul: /* Definition */

2013-08-03T01:40:11Z

Definition

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	===Explicit definition===		===Explicit definition===

	A [[group]] <math>G</math> is termed a '''Lazard Lie group''' ~~if there is a~~ natural number <math>c</math> ~~such that~~ '''both''' the following hold:		A [[group]] <math>G</math> is termed a class <math>c</math> '''Lazard Lie group''' for some natural number <math>c</math> if '''both''' the following hold:

	{\| class="sortable" border="1"		{\| class="sortable" border="1"
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	Condition (1) gets more demanding (i.e., stronger, so satisfied by fewer groups) as <math>c</math> increases, while condition (2) gets less demanding (i.e., weaker, so satisfied by fewer groups) as we increase <math>c</math>. Thus, a particular value of <math>c</math> may work for a group but larger and smaller values may not.		Condition (1) gets more demanding (i.e., stronger, so satisfied by fewer groups) as <math>c</math> increases, while condition (2) gets less demanding (i.e., weaker, so satisfied by fewer groups) as we increase <math>c</math>. Thus, a particular value of <math>c</math> may work for a group but larger and smaller values may not.

			A group is termed a Lazard Lie group if it is a class <math>c</math> Lazard Lie group for some natural number <math>c</math>.

	A Lazard Lie group is a group that can participate on the ''group'' side of the [[Lazard correspondence]]. The Lie ring on the other side is its [[Lazard Lie ring]].		A Lazard Lie group is a group that can participate on the ''group'' side of the [[Lazard correspondence]]. The Lie ring on the other side is its [[Lazard Lie ring]].

Vipul at 00:03, 3 August 2013

2013-08-03T00:03:36Z

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	\|-		\|-
	\| [[dissatisfies metaproperty::finite direct product-closed group property]] \|\| No \|\| [[Lazard Lie property is not finite direct product-closed]] \|\| It is possible to have Lazard Lie groups <math>G_1</math> and <math>G_2</math> such that the [[external direct product]] <math>G_1 \times G_2</math> is not a Lazard Lie group.		\| [[dissatisfies metaproperty::finite direct product-closed group property]] \|\| No \|\| [[Lazard Lie property is not finite direct product-closed]] \|\| It is possible to have Lazard Lie groups <math>G_1</math> and <math>G_2</math> such that the [[external direct product]] <math>G_1 \times G_2</math> is not a Lazard Lie group.
			\|}
	==Relation with other properties==		==Relation with other properties==

Vipul at 00:03, 3 August 2013

2013-08-03T00:03:12Z

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	A [[p-group]] is termed a '''Lazard Lie group''' if its [[defining ingredient::local nilpotency class\|3-local nilpotency class]] is at most <math>p - 1</math>. In other words, every subgroup of it generated by at most three elements has [[nilpotency class of a group\|nilpotency class]] at most <math>p - 1</math> where <math>p</math> is the prime associated with the group.		A [[p-group]] is termed a '''Lazard Lie group''' if its [[defining ingredient::local nilpotency class\|3-local nilpotency class]] is at most <math>p - 1</math>. In other words, every subgroup of it generated by at most three elements has [[nilpotency class of a group\|nilpotency class]] at most <math>p - 1</math> where <math>p</math> is the prime associated with the group.

			==Metaproperties==

			{\| class="sortable" border="1"
			! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
			\|-
			\| [[dissatisfies metaproperty::subgroup-closed group property]] \|\| No \|\| [[Lazard Lie property is not subgroup-closed]] \|\| It is possible to have a Lazard Lie group <math>G</math> and a subgroup <math>H</math> of <math>G</math> such that <math>H</math> is ''not'' a Lazard Lie group in its own right.
			\|-
			\| [[dissatisfies metaproperty::quotient-closed group property]] \|\| No \|\| [[Lazard Lie property is not quotient-closed]] \|\| It is possible to have a Lazard Lie group <math>G</math> and a [[normal subgroup]] <math>H</math> of <math>G</math> such that the [[quotient group]] <math>G/H</math> is ''not'' a Lazard Lie group in its own right.
			\|-
			\| [[dissatisfies metaproperty::finite direct product-closed group property]] \|\| No \|\| [[Lazard Lie property is not finite direct product-closed]] \|\| It is possible to have Lazard Lie groups <math>G_1</math> and <math>G_2</math> such that the [[external direct product]] <math>G_1 \times G_2</math> is not a Lazard Lie group.
	==Relation with other properties==		==Relation with other properties==

Vipul: /* Metaproperties */

2013-08-02T23:50:29Z

Metaproperties

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	* [[Stronger than::Locally nilpotent group]]		* [[Stronger than::Locally nilpotent group]]

	~~==Metaproperties==~~

	~~{{S-closed}}~~

	~~{{Q-closed}}~~

	~~{{finite-DP-closed}}~~

	~~===3-local===~~

	~~A group occurs as a Lazard Lie group if and only if, for any three elements of the group, the subgroup they generate occurs as a Lazard Lie group. {{proofat\|[[Lazard Lie property is 3-local]]}}~~

Vipul at 23:33, 2 August 2013

2013-08-02T23:33:39Z

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	A Lazard Lie group is a group that can participate on the ''group'' side of the [[Lazard correspondence]]. The Lie ring on the other side is its [[Lazard Lie ring]].		A Lazard Lie group is a group that can participate on the ''group'' side of the [[Lazard correspondence]]. The Lie ring on the other side is its [[Lazard Lie ring]].

			===Set of possible values <math>c</math> for which a group is a class <math>c</math> Lazard Lie group===

			A group is a Lazard Lie group if and only if its [[3-local nilpotency class]] is less than or equal to its [[powering threshold]]. The set of permissible <math>c</math> values for which the group is a class <math>c</math> Lazard Lie group is the set of <math>c</math> satisfying:

			3-local nilpotency class <math>\le c \le</math> powering threshold
	===p-group version===		===p-group version===

Vipul: /* Explicit definition */

2012-05-15T02:15:43Z

Explicit definition

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	! No. !! Shorthand for property !! Explanation		! No. !! Shorthand for property !! Explanation
	\|-		\|-
	\| 1 \|\| <math>G</math> is [[defining ingredient::powered group for a set of primes\|powered for the set]] of all primes less than or equal to <math>c</math>. \|\| <math>G</math> is uniquely <math>p</math>-divisible for all primes <math>p \le c</math>. In other words, if <math>p \le c</math> is a prime and <math>g \in G</math>, there is a unique value <math>h \in G</math> satisfying <math>h^p = g</math>.		\| 1 \|\| The [[powering threshold]] for <math>G</math> is at least <math>c</math>, i.e., <math>G</math> is [[defining ingredient::powered group for a set of primes\|powered for the set]] of all primes less than or equal to <math>c</math>. \|\| <math>G</math> is uniquely <math>p</math>-divisible for all primes <math>p \le c</math>. In other words, if <math>p \le c</math> is a prime and <math>g \in G</math>, there is a unique value <math>h \in G</math> satisfying <math>h^p = g</math>.
	\|-		\|-
	\| 2 \|\| The [[defining ingredient::local nilpotency class\|3-local nilpotency class]] of <math>G</math> is at most <math>c</math>. \|\| For any three elements of <math>G</math>, the subgroup of <math>G</math> generated by these three elements is a [[nilpotent group]] of [[nilpotency class]] at most <math>c</math>.		\| 2 \|\| The [[defining ingredient::local nilpotency class\|3-local nilpotency class]] of <math>G</math> is at most <math>c</math>. \|\| For any three elements of <math>G</math>, the subgroup of <math>G</math> generated by these three elements is a [[nilpotent group]] of [[nilpotency class]] at most <math>c</math>.

Vipul at 02:14, 15 May 2012

2012-05-15T02:14:48Z

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	==Definition==		==Definition==

			===Quick definition===

			A [[group]] is termed a '''Lazard Lie group''' if its [[defining ingredient::3-local nilpotency class]] is finite and less than or equal to the group's [[defining ingredient::powering threshold]].
			===Explicit definition===

	A [[group]] <math>G</math> is termed a '''Lazard Lie group''' if there is a natural number <math>c</math> such that '''both''' the following hold:		A [[group]] <math>G</math> is termed a '''Lazard Lie group''' if there is a natural number <math>c</math> such that '''both''' the following hold:

Vipul at 22:37, 4 May 2012

2012-05-04T22:37:12Z

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	{{~~prime-parametrized~~ group property}}		{{group property}}

	{{wikilocal}}		{{wikilocal}}

Vipul: /* Relation with other properties */

2011-02-10T20:11:05Z

Relation with other properties

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	===Stronger properties===		===Stronger properties===

	* [[Weaker than::~~Abelian~~ group]]		{\| class="sortable" border="1"
			! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
			\| [[Weaker than::abelian group]] \|\| any two elements commute \|\| Precisely the case <math>c = 1</math> (see also [[Lazard correspondence#Particular cases]]) \|\| \|\|{{intermediate notions short\|Lazard Lie group\|abelian group}}
			\|-
			\| [[Weaker than::Baer Lie group]] \|\| uniquely 2-divisible and class at most two \|\| Precisely the case <math>c = 2</math> (see also [[Lazard correspondence#Particular cases]] \|\| \|\| {{intermediate notions short\|Lazard Lie group\|Baer Lie group}}
			\|-
			\| [[Weaker than::p-group of nilpotency class less than p]] \|\| \|\| global nilpotency class puts an upper bound on the 3-local nilpotency class \|\| \|\| {{intermediate notions short\|Lazard Lie group\|p-group of nilpotency class less than p}}
			\|-
			\| [[Weaker than::rationally powered nilpotent group]] \|\| nilpotent and uniquely divisible for all primes \|\| \|\| \|\| {{intermediate notions short\|Lazard Lie group\|rationally powered nilpotent group}}
			\|}

	===Weaker properties===		===Weaker properties===