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
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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."
|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
|Property||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|
|Baer Lie group|
|LCS-Baer Lie group|