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Cyclic Subgroup Sum

We classify finite cyclic groups based on the sum of the orders of their proper subgroups Let $G$ be a finite cyclic group of order $n$, and let $x$ be a generator of $G$, so $G = \langle x \rangle$. Then $G \cong Z_n$, and by the Fundamental Theorem of Finite Cyclic Groups, we know that for each positive divisor $d$ of $n$, there is a unique cyclic subgroup of order $d$, namely $\langle x^{n/d} \rangle$.

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