Nilpotency is 2-local for finite groups

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