Schreier's Theorem

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Schreier's Refinement Theorem is a result in group theory. Otto Schreir discovered it in 1928, and used it to give an improved proof of the Jordan-Hölder Theorem. Six years later, Hans Zassenhaus published his lemma, which gives an improved proof of Schreier's Theorem.


Let $\Sigma_1$ and $\Sigma_2$ be composition series of a group $G$. Then there exist equivalent composition series $\Sigma'_1$ and $\Sigma'_2$ such that $\Sigma'_1$ is finer than $\Sigma_1$ and $\Sigma'_2$ is finer than $\Sigma_2$.


Suppose $\Sigma_1 = (H_i)_{0 \le i \le n}$ and $\Sigma_2 = (K_j)_{0\le j \le m}$ are the composition series in question. For integers $j \in [1,m-1]$, $i \in [0,n-1]$, let $H'_{im+j} = H_{i+1} \cdot (H_i \cap K_j)$, and for integers $i \in [0,n]$, let \[H'_{im} = H_i = H_{i+1} \cdot (H_i \cap K_0) = H_{i} \cdot (H_{i-1} \cap K_m),\] where these groups are defined. Similarly, for integers $i \in [1,n-1]$, $j\in [0,m-1]$, let $K'_{jn+i} = K_{j+1} \cdot (K_j \cap H_i)$, and for integers $j \in [0,m]$, define \[K'_{jn} = K_j = K_{j+1} \cdot (K_j \cap H_0) = K_j \cdot (K_{j-1} \cap H_n),\] where these groups are defined. Then by Zassenhaus's Lemma, $\Sigma'_1 = (H'_k)_{0 \le k \le mn}$ and $\Sigma'_2 = (K'_\ell)_{0 \le \ell \le mn}$ are composition series; they are evidently finer than $\Sigma_1$ and $\Sigma_2$, respectively. Again by Zassenhaus's Lemma, the quotients $H'_{im+j}/H'_{im+j+1}$ and $K'_{jn+i}/K'_{jn+i+1}$ are equivalent, so series $\Sigma'_1$ and $\Sigma'_2$ are equivalent, as desired. $\blacksquare$

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