Let $T$ be a first-order theory of cyclic groups. Even if an abelian group $(G,+)$ satisfy $(G,+)\models T$ there is no reason that $(G,+)$ is a cyclic. (For example, by Löwenheim–Skolem theorem there is uncountable abelian group $G$ that satisfy $T$.)
I tried to find a first-order formula that is true for all cyclic groups, but is false for some abelian group. But I don't know how to find it. Thanks for any help.