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 G satisfying the following two conditions:

  1. It is a nilpotent group of class two, i.e., its nilpotency class is at most two.
  2. It is a 2-powered group (also known as uniquely 2-divisible group): For every g \in G, there is a unique element h \in G such that h^2 = g.

Given condition (1), condition (2) is equivalent to requiring that G 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 G and a subgroup H of G such that H 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 G and a normal subgroup H of G such that the quotient group G/H 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 G_i, i \in I the external direct product of all the G_is 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