Baer Lie group
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This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: group of nilpotency class and uniquely 2-divisible group
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Definition
A Baer Lie group is a group satisfying the following two conditions:
- It is a nilpotent group of class two, i.e., its nilpotency class is at most two.
- It is a 2-powered group (also known as uniquely 2-divisible group): For every , there is a unique element such that .
Given condition (1), condition (2) is equivalent to requiring that be both 2-torsion-free (i.e., no element of order two) and 2-divisible. (see equivalence of definitions of nilpotent group that is torsion-free for a set of primes).
A Baer Lie group is a group that can serve on the group side of a Baer correspondence, i.e., it has a corresponding Baer Lie ring.
A finite group is a Baer Lie group if and only if it is an odd-order class two group.
Metaproperties
Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
subgroup-closed group property | No | Baer Lie property is not subgroup-closed | It is possible to have a Baer Lie group and a subgroup of such that is not a Baer Lie group in its own right. |
quotient-closed group property | No | Baer Lie property is not quotient-closed | It is possible to have a Baer Lie group and a normal subgroup of such that the quotient group is not a Baer Lie group in its own right. |
direct product-closed group property | Yes | Baer Lie property is direct product-closed | Given Baer Lie groups the external direct product of all the s is also a Baer Lie group. |
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
odd-order abelian group | Odd-order class two group|FULL LIST, MORE INFO | |||
odd-order class two group | group of odd order and nilpotency class two; equivalently, a finite Baer Lie group. | |FULL LIST, MORE INFO | ||
rationally powered class two group | |FULL LIST, MORE INFO |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
Lazard Lie group | Powering threshold is greater than or equal to 3-local nilpotency class | Global Lazard Lie group|FULL LIST, MORE INFO | ||
global Lazard Lie group | Powering threshold is greater than or equal to nilpotency class | |FULL LIST, MORE INFO | ||
UCS-Baer Lie group | class two, and center is 2-powered | |FULL LIST, MORE INFO | ||
LCS-Baer Lie group | class two, and derived subgroup is 2-powered | |FULL LIST, MORE INFO | ||
LUCS-Baer Lie group | class two, and every element in derived subgroup has unique square root in center | UCS-Baer Lie group|FULL LIST, MORE INFO |