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Abstract Algebra

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Cyclic Group: When Is a Finite Abelian Group cyclic?

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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?

A

There is an element $g\in G$ of order $|G|$.

B

$G$ is a direct product of two cyclic groups of order $p\ge 2$ and $q\ge2$, where $p$ and $q$ are coprime.

C

For every $d\ge 1$, the number of elements of $G$ with $dg=e$ is at most $d$.

D

$G$ is a direct product of two cyclic groups of order $p\ge2$ and $q\ge2$, where $p$ divides $q$.