I've recently become interested in the elementary theory of groups due to Sela and Myasnikov-Kharlampovich's work with free groups. I'd like a good introduction to the field of the elementary theory of groups, and in particular I'd like a reference to contain examples of group properties that cannot be read from a group's elementary theory. For example, it seems that the statements "G is vritually abelian" or "G is hopfian" could not be expressed with first-order sentences, but I don't have enough knowledge yet to determine if such things are true. Does anyone know such a reference? Or, specifically, does someone know a proof of the fact that being hopfian (or some other group property) can't be read from a group's elementary theory? Thanks!



need a good reference for introduction to elementary theory of groups
I learned a lot from reading Bestvina and Feighn's article [Notes on Sela's work: Limit groups and Makanin-Razborov diagrams](http://arxiv.org/abs/0809.0467). It's not a broad introduction to elementary theory, but it does express some of Sela's ideas quite succinctly. You may need a background in geometric group theory (specifically, in understanding laminations on 2-complexes or group actions on $\mathbb{R}$-trees) to get the most out of it.


Regarding the question of whether or not the Hopf property is elementary: the answer is obviously `no' if you allow infinitely generated examples. Indeed, consider the free group on countably many generators, $F\_\infty$. The elementary theory is completely determined by the set of finitely generated subgroups, so $F\_\infty$ is elementary equivalent to $F\_2$. But $F\_2$ is Hopfian and $F\_\infty$ is not.


**EDIT:** I'm getting a little nervous about the claim that the elementary theory is determined by the list of finitely generated subgroups. However, Sela and Kharlampovich--Miasnikov proved that the natural inclusions $F\_n\subseteq F\_{n+1}$ are elementary embeddings, from which it does indeed follow that $F\_\infty$ is elementarily equivalent to $F\_2$.


I don't know a finitely generated example, although I agree that one must surely exist.


