# LCS-Lazard Lie group

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Definition

A **LCS-Lazard Lie group** is a group satisfying **both** the following properties:

- It is a 3-locally nilpotent group, i.e., any three elements of the group generate a nilpotent subgroup.
- Its 3-local lower central series powering threshold is . Explicitly, for any nonnegative integer , let denote the member of the 3-local lower central series of . Then, is powered over all the primes .

The definition of LCS-Lazard Lie group is somewhat nicer than the definition of Lazard Lie group in that it does not involve a "nilpotency class-specific definition."

## Metaproperties

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

subgroup-closed group property | No | LCS-Lazard Lie property is not subgroup-closed | It is possible to have a LCS-Lazard Lie group and a subgroup of such that is not a Lazard Lie group. |

quotient-closed group property | No | LCS-Lazard Lie property is not quotient-closed | It is possible to have a LCS-Lazard Lie group and a normal subgroup of such that the quotient group is not a LCS-Lazard Lie group. |

finite direct product-closed group property | Yes | LCS-Lazard Lie property is finite direct product-closed | If and are LCS-Lazard Lie groups, then the external direct product is a LCS-Lazard Lie group. |

## Relation with other properties

### Stronger properties

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

abelian group | ||||

Baer Lie group | ||||

LCS-Baer Lie group |