On Sunday, July 14, 2013 5:15:13 AM UTC-5, peps...@gmail.com wrote: I think I get it now. When a non-abelian group has an abelian subgroup, this is a non-elementary extension because the statement "For all x, For all y, xy = yx" is false in the larger group but true in the subgroup. I think the question with which you opened the post is basically equivalent to asking whether the theorem cited by David is correct. Am I more on target now?
Closer but not there yet. Davild cited a result saying that all free groups (on two or more generators, I guess) are elementarily equivalent, which is apparently weaker than saying that one is an elementary extension of the other.
Two structures A and B are "elementarily equivalent" if they satisfy the same first order *sentences* (formulas without free variables); thus, if A is an Abelian group, so is B. A is an "elementary extension" of B (in other words B is an "elementary substructure" of A" if, for any first order *formula* phi (which may have free variables), and any assignment of values in B to the free variables, phi is satisfied in B if and only if it's satisfied in A.
For example, consider the group A = (Z,+) of all integers, and the subgroup B = (2Z,+) of the even integers. B is isomorphic (and a fortiori elementarily equivalent) to B, but A is not an elementary extension of B. To see this, consider the formula phi(x) := (exist y)(y + y = x), and observe that phi(2) holds in B but does not hold in A.