# Difference between revisions of "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: 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.

## 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 group of odd order and nilpotency class two; equivalently, a finite Baer Lie group.
rationally powered class two group

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Lazard Lie group