Lemma 2

Let $a$ be an integer $> 0$. By associating with every composition series $(\mathbf{H}_i)_{0 \leq i \leq n}$ of the group $\mathbf{Z}/a\mathbf{Z}$ the sequence $(s_i)_{1 \leq i \leq n}$, where $s_i$ is the order of $\mathbf{H}_{i-1}/\mathbf{H}_i$, a one-to-one correspondence is obtained between the composition series of $\mathbf{Z}/a\mathbf{Z}$ and the finite sequences $(s_i)$ of integers $> 0$ such that $a = s_1 \ldots s_n$. The composition series $(\mathbf{H}_i)_{0 \leq i \leq n}$ is a Jordan-Hölder series if and only if the $s_i$ are prime.

If $(\mathbf{H}_i)_{0 \leq i \leq n}$ is a composition series of $\mathbf{Z}/a\mathbf{Z}$, it follows, by induction on $n$, from no. 4, Proposition 4, that $a = \prod_{i=1}^n (\mathbf{H}_{i-1} : \mathbf{H}_i)$.

Conversely, let $(s_i)_{1 \leq i \leq n}$ be a sequence of integers $> 0$ such that $a = s_1 \ldots s_n$. If $(\mathbf{H}_i)_{0 \leq i \leq n}$ is a composition series of $\mathbf{Z}/a\mathbf{Z}$ such that $(\mathbf{H}_{i-1} : \mathbf{H}_i) = s_i$ for $1 \leq i \leq n$, then necessarily $((\mathbf{Z}/a\mathbf{Z}) : \mathbf{H}_i) = \prod_{1 \leq j \leq i} s_j$ as is seen by induction on $i$, whence $\mathbf{H}_i = \left( \prod_{j=1}^i s_j \right) \mathbf{Z}/a\mathbf{Z}$ (Proposition 19), which shows the injectivity of the mapping in question. We now show its surjectivity. Writing $\mathbf{H}_i = \left( \prod_{j=1}^i s_j \right) \mathbf{Z}/a\mathbf{Z}$ for $0 \leq i \leq n$, a composition series of $\mathbf{Z}/a\mathbf{Z}$ is obtained such that $(\mathbf{H}_{i-1} : \mathbf{H}_i) = s_i$ (Proposition 19). The second assertion follows from Proposition 20 and no. 7, Proposition 9.

Let $\mathfrak{P}$ denote the set of prime numbers.