# Difference between revisions of "Baer Lie group"

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# It is a [[defining ingredient::nilpotent group]] of [[defining ingredient::group of nilpotency class two|class two]], i.e., its [[defining ingredient::nilpotency class]] is at most two. | # It is a [[defining ingredient::nilpotent group]] of [[defining ingredient::group of nilpotency class two|class two]], i.e., its [[defining ingredient::nilpotency class]] is at most two. | ||

− | # It is a [[defining ingredient:: | + | # It is a [[defining ingredient::2-powered group]]: For every <math>g \in G</math>, there is a unique element <math>h \in G</math> such that <math>h^2 = g</math>. |

− | Given condition (1), condition (2) is equivalent to requiring that <math>G</math> be both [[torsion-free group|torsion-free]] (i.e., | + | Given condition (1), condition (2) is equivalent to requiring that <math>G</math> be both [[2-torsion-free group|2-torsion-free]] (i.e., no element of order two) and [[2-divisible group|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 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]]. |

## Revision as of 15:43, 5 August 2013

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

View other group property conjunctions OR view all group properties

## 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 |
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odd-order abelian group | ||||

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

### Weaker properties

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