Finite Lazard 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: finite group and Lazard Lie group
View other group property conjunctions OR view all group properties

Definition

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

No. Shorthand Explanation
1 finite and a Lazard Lie group G is a finite group and there is a natural number c such that the 3-local nilpotency class of G is at most c and the order of G does not have any prime divisors less than or equal to c.
2 product of its Sylow subgroups, each of which is a Lazard Lie group G 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 p-group is a Lazard Lie group if its 3-local nilpotency class is at most p.

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 c = 1 |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 c = 2 |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