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Let $G$ be a finite abelian group with identity (neutral) element $e$. We use the additive notation for the group law.
Which of the following does not imply that $G$ is cyclic?
There is an element $g\in G$ of order $|G|$.
$G$ is a direct product of two cyclic groups of order $p\ge 2$ and $q\ge2$, where $p$ and $q$ are coprime.
For every $d\ge 1$, the number of elements of $G$ with $dg=e$ is at most $d$.
$G$ is a direct product of two cyclic groups of order $p\ge2$ and $q\ge2$, where $p$ divides $q$.