An equivalent characterization of finite cyclic groups

Prove the following proposition by invoking the Fundamental Theorem of Finite Abelian Groups.

Proposition: Let G be a finite abelian group such that, for all positive integers n dividing |G|, G contains at most n elements satisfying x^n = 1. Prove that G is cyclic.

Proof: By FTFGAG, we have G \cong Z_{n_1} \times \cdots \times Z_{n_t} for some integers n_i such that n_{i+1}|n_i for all i. Suppose t \geq 2, and let p be a prime divisor of n_2. Now p also divides n_1. Now Z_{n_2} contains an element x of order p, and also Z_{n_1} contains an element y of order p. Now every element z \in \langle y \rangle \times \langle x \rangle satisfies z^p = 1, and there are p^2 such elements, a contradiction. Thus t = 1, hence G = Z_{n_1} is cyclic.

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