Abelianness is 2-local

Statement with symbols
A group $$G$$ is Abelian if and only if the subgroup $$\langle a,b \rangle$$ is Abelian for any two elements $$a,b \in G$$.

Applications

 * Abelianness is directed union-closed: This follows from the fact that any local property is directed union-closed.

Other related facts
Some other results about being local:


 * Engel is 2-local
 * Nilpotence class c is (c plus 1)-local
 * Nilpotence class two is 3-local
 * Solvable length l is 2^l-local

On the other hand, the property of being cyclic is not local; there are locally cyclic groups that are not cyclic.

Related facts for finite groups

 * Cyclicity is 2-local for finite groups
 * Solvability is 2-local for finite groups

Abelian implies the subgroup generated by any two elements is Abelian
This follows from the fact that Abelianness is subgroup-closed: any subgroup of an Abelian group is Abelian.

Subgroup generated by any two elements is Abelian implies Abelian
The key idea here is that the condition for Abelianness is a universal identity that has only two free variables.

Given: A group $$G$$ such that the subgroup generated by any two elements of $$G$$ is Abelian.

To prove: $$G$$ is Abelian: for any $$a,b \in G$$, we have $$ab = ba$$.

Proof: Consider the subgroup generated by $$a$$ and $$b$$. By assumption, this is Abelian, and since $$a,b$$ are both elements of this subgroup, we obtain $$ab = ba$$.