2-locally nilpotent group
Definition
A group is termed a 2-locally nilpotent group if every 2-generated subgroup of it is a nilpotent group.
The 2-local nilpotency class of a 2-locally nilpotent group is defined as the supremum, over all 2-generated subgroups, of their nilpotency class. The 2-local nilpotency class of a 2-locally nilpotent group may be infinite. An example is the generalized dihedral group for 2-quasicyclic group.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| nilpotent group | |FULL LIST, MORE INFO | |||
| locally nilpotent group | every finitely generated subgroup is nilpotent | |FULL LIST, MORE INFO | ||
| 3-locally nilpotent group | every subgroup generated by three elements is nilpotent | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Engel group | |FULL LIST, MORE INFO |