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

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:

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

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 $ab=ba$ .