# Abelianness is 2-local

From Groupprops

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)

View all group metaproperty satisfactions | View all group metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for group properties

Get more facts about Abelian group |Get facts that use property satisfaction of Abelian group | Get facts that use property satisfaction of Abelian group|Get more facts about 2-local group property

## Contents

## Statement

### Statement with symbols

A group is Abelian if and only if the subgroup is Abelian for any two elements .

## Related facts

### Applications

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

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

## 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 such that the subgroup generated by any two elements of is Abelian.

**To prove**: is Abelian: for any , we have .

**Proof**: Consider the subgroup generated by and . By assumption, this is Abelian, and since are both elements of this subgroup, we obtain .