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The question is as follows:

Let $G$ be a finite and abelian group that satisfies $$\#\{g\in G\,\vert\, g^p=e\}\leq p$$ for all $p\in\mathbb{N}$, where $e$ is the identity element in $G$. Prove that $G$ is a cyclic group.

It doesn't seem difficult, but I have hit a wall and I don't know where to start. Any help or hit would be highly appreciated!

Álvaro Serrano
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Hint: If $G$ is not cyclic, you can use the classification of finite abelian groups to show that there is some prime $p$ such that $G$ has a subgroup isomorphic to $\Bbb Z_p\times \Bbb Z_p$.

Arthur
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