# Abelianness is 2-local

This article gives the statement, and possibly proof, of a group property (i.e., Abelian group) satisfying a group metaproperty (i.e., 2-local group property)
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## Statement

### Statement with symbols

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

## Related facts

### Other related facts

Some other results about being local:

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

## Proof

### 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 $ab = ba$.