Theorem 5 (Schreier)

*Given two composition series $\Sigma_1, \Sigma_2$ of a group with operators G, there exist two equivalent composition series $\Sigma'_1, \Sigma'_2$, finer respectively than $\Sigma_1$ and $\Sigma_2$.

Let $\Sigma_1 = (H_i)_{0 \leq i \leq n}$ and $\Sigma_2 = (K_j)_{0 \leq j \leq p}$ be the two given composition series with respectively $n + 1$ and $p + 1$ terms; we shall see that the composition series $\Sigma'_1$ can be formed by inserting $p - 1$ subgroups $H'_{i,j}$ (1 \leq j \leq p - 1) between H_i and H_{i+1} for 0 \leq i \leq n - 1 and the series \Sigma_2' by inserting n - 1 subgroups K_{j,i}' (1 \leq i \leq n - 1) between K_j and K_{j+1} for 0 \leq j \leq p - 1; thus two series of pn + 1 stable subgroups of G will be obtained; by choosing suitably the inserted stable subgroups, we shall show that these series are equivalent composition series.

To this end note that H_i \cap K_j is a stable subgroup of H_i and of K_j, and hence (Theorem 4) H_{i+1}.(H_i \cap K_j) is a stable subgroup of H_i containing H_{i+1} and K_{j+1}.(H_i \cap K_j) is a stable subgroup of K_j containing K_{j+1}. If we write H_{i,j}' = H_{i+1}.(H_i \cap K_j) and K_{j,i}' = K_{j+1}.(H_i \cap K_j), H_{i,j+1}' is a stable subgroup of H_{i,j}' (0 \leq j \leq p - 1) and K_{j,i+1}' is a stable subgroup of K_{j,i}' (0 \leq i \leq n - 1). Moreover H_{i,0}' = H_i, H_{i,p}' = H_{i+1}, K_{j,0}' = K_j and K_{j,p}' = K_{j+1}. To show the theorem, it suffices to show that H_{i,j+1}' (resp. K_{j,i+1}') is a normal stable subgroup of H_{i,j}' (resp. K_{j,i}') and that the quotient groups H_{i,j}'/H_{i,j+1}' and K_{j,i}'/K_{j,i+1}' are isomorphic (0 \leq i \leq n - 1, 0 \leq j \leq p - 1). This follows from the following lemma by taking H = H_i, H' = H_{i+1}, K = K_j, K' = K_{j+1}.