Nilpotency is 2-local for finite groups

Statement

Verbal statement

The following are equivalent for a finite group:

• The group is nilpotent, i.e., it is a finite nilpotent group.
• The subgroup generated by any two elements of the group is a nilpotent group.

Statement with symbols

Let $G$ be a finite group. The following are equivalent for $G$:

• $G$ is nilpotent.
• For any $a,b\in G$, the subgroup $\langle a,b\rangle$ is nilpotent.

Related facts

• Abelianness is 2-local
• Cyclicity is 2-local for finite groups
• Solvability is 2-local for finite groups