Proving the Fundamental Theorem of Arithmetic via Jordan–Hölder

Let \(n \in \mathbb{Z}\). By Jordan–Hölder, the cyclic group \(Z_n\) has a composition series \(1 = G_0 \trianglelefteq G_1 \trianglelefteq \ldots \trianglelefteq G_s = Z_n\), where \(G_{i+1}/G_i\) is simple. It is known that quotient groups and subgroups of cyclic groups are simple. Thus all \(G_i\), and hence all \(G_{i+1}/G_i\) are cyclic. In particular, \(G_{i+1}/G_i\) must have prime order \(p_i\) due to its simplicity. So we can conclude that \(n = \left|Z_n\right| = \left|G_s\right| = \left|G_s/G_{s-1}\right|\left|G_{s-1}/G_{s-2}\right| \ldots \left|G_1/G_0\right| = p_{s-1}p_{s-2} \ldots p_0\) is a product of primes. Furthermore, the composition series is unique up to isomorphism of quotient groups and rearrangement, so the list of \(p_i\)'s is unique. Thus \(n\) has a unique prime factorization.