TITLE: Showing that the Class of Cyclic Groups Aren't Axiomatizable
QUESTION [6 upvotes]: The class of finite cyclic groups are not axiomatizable, for if we supposed they were by some set of sentences $\Sigma$, then there would exist a model for $\Sigma$ of at least order $n$ for all $n \in \mathbb{N}$ (i.e., the model $Z_{n+1}$ for each $n$).  Then we would have that there would also exist an infinite model as well by the compactness theorem (and hence we would have a contradiction).
But evidently the theory all cyclic groups (including the infinite ones) are also not axiomatizable.  Why is this so?

REPLY [3 votes]: André's answer provides an answer to your question, but the result can be strengthened: the cyclic groups are not axiomatisable even among countable groups. 
This is, in part, because the theory of $({\bf Z},+)$ is not $\omega$-categorical. For instance ${\bf Z}\oplus {\bf Q}$ is elementarily equivalent to it, and is evidently not cyclic. In fact, the theory has $2^{\aleph_0}$ nonisomorphic countable models. To see this, notice that there are that many complete types over $\emptyset$.
On the other hand, using the classification theorem for finitely generated abelian groups, one can show that cyclic groups are elementary relative to the (clearly non-elementary) class of finitely generated groups.