LCS-Lazard Lie group

From Groupprops
Jump to: navigation, search
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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 properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions


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

  1. It is a 3-locally nilpotent group, i.e., any three elements of the group generate a nilpotent subgroup.
  2. Its 3-local lower central series powering threshold is \infty. Explicitly, for any nonnegative integer k, let \gamma_k^{3-loc}(G) denote the k^{th} member of the 3-local lower central series of G. Then, \gamma_k^{3-loc}(G) is powered over all the primes p \le k.

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."


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 G and a subgroup H of G such that H 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 G and a normal subgroup H of G such that the quotient group G/H is not a LCS-Lazard Lie group.
finite direct product-closed group property Yes LCS-Lazard Lie property is finite direct product-closed If G_1 and G_2 are LCS-Lazard Lie groups, then the external direct product G_1 \times G_2 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