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Groupprops β

Lie ring of nilpotency class two


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