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Schreier's Theorem

From AoPSWiki

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.

Statement

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 $\Sigma_2$ .

Proof

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. $\blacksquare$

See also

Jordan-Hölder Theorem

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Category: Group theory

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