## 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.