Lazard Lie group: Difference between revisions

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Definition

Quick definition

A group is termed a Lazard Lie group if its 3-local nilpotency class is finite and less than or equal to the group's powering threshold.

Explicit definition

A group $G$ is termed a class $c$ Lazard Lie group for some natural number $c$ if both the following hold:

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

Condition (1) gets more demanding (i.e., stronger, so satisfied by fewer groups) as $c$ increases, while condition (2) gets less demanding (i.e., weaker, so satisfied by more groups) as we increase $c$ . Thus, a particular value of $c$ 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 $c$ Lazard Lie group for some natural number $c$ .

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 $c$ for which a group is a class $c$ 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 $c$ values for which the group is a class $c$ Lazard Lie group is the set of $c$ satisfying:

3-local nilpotency class $\leq c\leq$ powering threshold

p-group version

A p-group is termed a Lazard Lie group if its 3-local nilpotency class is at most $p-1$ . In other words, every subgroup of it generated by at most three elements has nilpotency class at most $p-1$ where $p$ is the prime associated with the group.

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
subgroup-closed group property	No	Lazard Lie property is not subgroup-closed	It is possible to have a Lazard Lie group $G$ and a subgroup $H$ of $G$ such that $H$ is not a Lazard Lie group in its own right.
quotient-closed group property	No	Lazard Lie property is not quotient-closed	It is possible to have a Lazard Lie group $G$ and a normal subgroup $H$ of $G$ such that the quotient group $G/H$ is not a Lazard Lie group in its own right.
finite direct product-closed group property	No	Lazard Lie property is not finite direct product-closed	It is possible to have Lazard Lie groups $G_{1}$ and $G_{2}$ such that the external direct product $G_{1}\times G_{2}$ is not a Lazard Lie group.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Intermediate notions
abelian group	any two elements commute	Precisely the case $c=1$ (see also Lazard correspondence#Particular cases)	\|FULL LIST, MORE INFO
Baer Lie group	uniquely 2-divisible and class at most two	Precisely the case $c=2$ (see also Lazard correspondence#Particular cases	\|FULL LIST, MORE INFO
p-group of nilpotency class less than p		global nilpotency class puts an upper bound on the 3-local nilpotency class	\|FULL LIST, MORE INFO
rationally powered nilpotent group	nilpotent and uniquely divisible for all primes		\|FULL LIST, MORE INFO

Weaker properties

Locally nilpotent group

@@ Line 1: / Line 1: @@
-{{prime-parametrized group property}}
+{{group property}}
 {{wikilocal}}
@@ Line 5: / Line 5: @@
 ==Definition==
-A [[p-group]] is termed a '''Lazard Lie group''' if every subgroup of it generated by 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.
+===Quick definition===
-==Relation with other properties==
+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 class <math>c</math> '''Lazard Lie group''' for some natural number <math>c</math> if '''both''' the following hold:
+{| class="sortable" border="1"
+! No. !! Shorthand for property !! Explanation
+|-
+| 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>.
+|}
+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 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]].
-===Stronger properties===
+===Set of possible values <math>c</math> for which a group is a class <math>c</math> Lazard Lie group===
-* [[Weaker than::p-group of nilpotency class less than p]]
+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:
-* [[Weaker than::Abelian p-group]]
-==Facts==
+-local nilpotency class <math>\le c \le</math> powering threshold
+===p-group version===
-[[Lazard's theorem]] gives a method to construct a Lazard Lie ring for any Lazard Lie group. This construction and its paraphernalia go under the name of the [[Lazard correspondence]].
+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==
-{{S-closed}}
+{| 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==
-{{Q-closed}}
+===Stronger properties===
-{{finite-DP-closed}}
+{| 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}}
+|}
-===3-local===
+===Weaker properties===
-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]]}}
+* [[Stronger than::Locally nilpotent group]]