Local nilpotency class
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
Suppose is a group and is a natural number. The -local nilpotency class is defined as the supremum, over all subgroups of generated by sets of size at most , of the nilpotency class of . In other words, it is defined as:
If there is a non-nilpotent subgroup of generated by or fewer elements, then the -local nilpotency class is . The -local nilpotency class may also be infinite because, even though all the subgroups generated by at most elements are nilpotent, there is no finite upper bound on their nilpotency class.
- 2-local nilpotency class is significant because many of the formulas and constructions for nilpotent groups involve two variables.
- 3-local nilpotency class is significant because the variety of groups is itself 3-local, and most correspondences, such as the Lazard correspondence, rely only on the 3-local behavior.
- The -local nilpotency class of a nontrivial group is always . This is because cyclic implies abelian.
- For , the -nilpotency class is less than or equal to the -nilpotency class.
- For any nilpotent group and any , the -local nilpotency class of a group is bounded by the nilpotency class of the group.
- If the -local nilpotency class of a group is a value , then the whole group is nilpotent of class . In other words, nilpotency of class is -local. For instance, abelianness is 2-local, and nilpotency of class two is 3-local.