# Elementary equivalence of free groups

This must be known inside out by model theorists by I have no cluse whether the following is true or not:

Denote by $F_n$ the free group on $n$ generators. Suppose that $n\neq m$. Are the groups $F_n$ and $F_m$ elementarily equivalent? What if we allow the set of generators to be countably infinite?

I'd appreciate any hints for the proof of the answer to this question. Looking at ultrapowers seem to me to be intractable, but who knows?

• Note that we want to ask when $m,n \ge 2$, since $m=1$ can be distinguished by abelianness. Jul 16 '14 at 19:16

• It includes free groups on infinitely many generators. Though for them it is much easier. For infinite $X \subseteq Y$ it is easy to show that $F_X \preceq F_Y$ by Tarski-Vaught test. Jul 17 '14 at 9:24
• @LevonHaykazyan Do you know whether 1) $F_X \equiv F_Y$ when $2 < |X| < \aleph_0$ and $Y$ is infinite? 2) $F_X\preceq F_Y$ when $X\subseteq Y$ and $X$ and $Y$ are finite? Of course, a positive answer to (2) implies a positive answer to (1)... Jul 19 '14 at 1:37