# Finite Lazard Lie group

From Groupprops

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: finite group and Lazard Lie group

View other group property conjunctions OR view all group properties

## Contents

## Definition

A **finite Lazard Lie group** is defined as a group satisfying the following **equivalent** conditions:

No. | Shorthand | Explanation |
---|---|---|

1 | finite and a Lazard Lie group | is a finite group and there is a natural number such that the 3-local nilpotency class of is at most and the order of does not have any prime divisors less than or equal to . |

2 | product of its Sylow subgroups, each of which is a Lazard Lie group | is a finite nilpotent group, i.e., it is the internal direct product of its Sylow subgroups, and each of its Sylow subgroups is a Lazard Lie group. Here, a finite -group is a Lazard Lie group if its 3-local nilpotency class is at most . |

## Relation with other properties

### Stronger properties

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

finite abelian group | any two elements commute | Special case | |FULL LIST, MORE INFO | |

odd-order class two group (also called finite Baer Lie group) | odd-order group of class at most two | Special case | |FULL LIST, MORE INFO |

### Weaker properties

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

finite nilpotent group | Finite group that is 1-isomorphic to an abelian group|FULL LIST, MORE INFO |