# Difference between revisions of "Baer Lie group"

From Groupprops

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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ||

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− | | [[Weaker than::odd-order abelian group]] || || || || | + | | [[Weaker than::odd-order abelian group]] || || || || {{intermediate notions short|Baer Lie group|odd-order abelian group}} |

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− | | [[Weaker than::odd-order class two group]] || group of odd order and nilpotency class two; equivalently, a ''finite'' Baer Lie group. || || || | + | | [[Weaker than::odd-order class two group]] || group of odd order and nilpotency class two; equivalently, a ''finite'' Baer Lie group. || || || {{intermediate notions short|Baer Lie group|odd-order class two group}} |

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− | | [[Weaker than::rationally powered class two group]] || || || || | + | | [[Weaker than::rationally powered class two group]] || || || || {{intermediate notions short|Baer Lie group|rationally powered class two group}} |

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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||

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− | | [[Stronger than::Lazard Lie group]] || || || || | + | | [[Stronger than::Lazard Lie group]] || || || || {{intermediate notions short|Lazard Lie group|Baer Lie group}} |

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+ | | [[Stronger than::global Lazard Lie group]] || || || || {{intermediate notions short|global Lazard Lie group|Baer Lie group}} | ||

+ | |- | ||

+ | | [[Stronger than::UCS-Baer Lie group]] || || || || {{intermediate notions short|UCS-Baer Lie group|Baer Lie group}} | ||

+ | |- | ||

+ | | [[Stronger than::LCS-Baer Lie group]] || || || || {{intermediate notions short|LCS-Baer Lie group|Baer Lie group}} | ||

+ | |- | ||

+ | | [[Stronger than::LUCS-Baer Lie group]] || || || || {{intermediate notions short|LUCS-Baer Lie group|Baer Lie group}} | ||

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## Revision as of 04:19, 6 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 |
---|---|---|---|---|

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 |