2-locally nilpotent group

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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 3-locally nilpotent group, Locally nilpotent group|FULL LIST, MORE INFO
locally nilpotent group every finitely generated subgroup is nilpotent 3-locally nilpotent group|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