# Baer Lie group

## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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 $G_i$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