Let $\mathfrak{A}$ be the class of all Abelian groups. It is easy to show that there is no first-order sentence $\varphi$ that can distinguish cyclic and non-cyclic groups in this setting.
Consider the group $(\mathbb{Z}, +)$, which is cyclic. By upward Löwenheim-Skolem, there exists a group $G$ with cardinality $\beth_1$ that is elementarily equivalent to $(\mathbb{Z}, +)$. However, $G$ cannot be cyclic since any group with exactly one generator is countable.
Suppose, then, that we consider $\mathfrak{B}$, the class of all finite Abelian groups. Is there a sentence in the language of groups for $\mathfrak{B}$ that distinguishes cyclic and non-cyclic groups?