Schreier refinement theorem

Remarks

Let the two series be $(G_i)_{0 \le i \le n}$ and $(H_j)_{0 \le j \le m}$. The proof inserts the groups $G_{i,j} = G_{i-1}(G_i \cap H_j)$ between $G_{i-1}$ and $G_i$, and the groups $H_{j,i} = H_{j-1}(H_j \cap G_i)$ between $H_{j-1}$ and $H_j$. Both refined series have length $nm$, and the Zassenhaus lemma gives $G_{i,j}/G_{i,j-1} \cong H_{j,i}/H_{j,i-1}$ for each pair $(i, j)$. For example, in $\Z/12\Z$ the series $0 \le 6\Z/12\Z \le \Z/12\Z$ and $0 \le 4\Z/12\Z \le \Z/12\Z$ refine to $0 \le 6\Z/12\Z \le 2\Z/12\Z \le \Z/12\Z$ and $0 \le 4\Z/12\Z \le 2\Z/12\Z \le \Z/12\Z$. Both refinements have factors of orders $2, 3, 2$ in some order, namely $\Z/2\Z$, $\Z/3\Z$, $\Z/2\Z$ and $\Z/3\Z$, $\Z/2\Z$, $\Z/2\Z$.

Going deeper

For a student

The theorem is the main step in the proof of the Jordan–Hölder theorem. A composition series has no proper refinement other than itself with repeated terms, so two composition series are already equivalent once the repeated terms are removed. The same argument works for groups with operators and for modules, where it gives the uniqueness of composition factors of modules of finite length.