Lie ring of nilpotency class two

From Groupprops
Revision as of 23:43, 10 February 2011 by Vipul (talk | contribs) (Weaker properties)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
ANALOGY: This is an analogue in Lie ring of a property encountered in group. Specifically, it is a Lie ring property analogous to the group property: group of nilpotency class two
View other analogues of group of nilpotency class two | View other analogues in Lie rings of group properties (OR, View as a tabulated list)

Definition

QUICK PHRASES: class two, inner derivation Lie ring is abelian, derived subring inside center, Lie brackets are central, triple Lie brackets are trivial

Symbol-free definition

A group is said to be of nilpotency class two or nilpotence class two if it satisfies the following equivalent conditions:

  1. Its nilpotency class is at most two, i.e., it is nilpotent of class at most two.
  2. Its derived subring (i.e., the subring generated by all elements arising from the image of the Lie bracket mapping) is contained in its center
  3. The commutator of any two elements of the group is central
  4. Any triple commutator (i.e., a commutator where one of the terms is itself a commutator) gives the zero element
  5. Its Lie ring of inner derivations is abelian.
NOTE: nilpotency class two is occasionally used to refer to a Lie ring whose nilpotency class is precisely two, i.e., a non-abelian Lie ring whose nilpotency class is two. This is a more restrictive use of the term than the typical usage, which includes abelian Lie rings.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
abelian Lie ring Lie bracket is trivial LCS-Baer Lie ring, Lie ring arising as the double of a class two Lie cring, Lie ring arising as the skew of a class two near-Lie cring, Lie ring whose bracket is the double of a Lie bracket giving nilpotency class two|FULL LIST, MORE INFO
Baer Lie ring Lie ring of nilpotency class two where every element has a unique half LCS-Baer Lie ring, LUCS-Baer Lie ring, Lie ring arising as the double of a class two Lie cring, Lie ring arising as the skew of a class two near-Lie cring, Lie ring whose bracket is the double of a Lie bracket giving nilpotency class two|FULL LIST, MORE INFO
Lie ring whose bracket is the double of a Lie bracket giving nilpotency class two Lie ring arising as the double of a class two Lie cring, Lie ring arising as the skew of a class two near-Lie cring|FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
nilpotent Lie ring |FULL LIST, MORE INFO
metabelian Lie ring derived subring is abelian 2-Engel Lie ring, Lie ring of nilpotency class three|FULL LIST, MORE INFO
2-Engel Lie ring Any triple commutator where two of the three inputs are equal must be trivial, i.e., [[x,y],x] = 0 for all x,y. |FULL LIST, MORE INFO