# 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: 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.

## 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 | Global Lazard Lie group|FULL LIST, MORE INFO | |||

global Lazard Lie group | |FULL LIST, MORE INFO | |||

UCS-Baer Lie group | |FULL LIST, MORE INFO | |||

LCS-Baer Lie group | |FULL LIST, MORE INFO | |||

LUCS-Baer Lie group | UCS-Baer Lie group|FULL LIST, MORE INFO |