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