# Difference between revisions of "Schreier's Theorem"

(New page: '''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 year...) |
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== Proof == | == Proof == | ||

− | Suppose <math>\Sigma_1 = (H_i)_{0 \le i \le n | + | Suppose <math>\Sigma_1 = (H_i)_{0 \le i \le n}</math> and <math>\Sigma_2 = (K_j)_{0\le j \le m}</math> are the composition series in question. For integers <math>j \in [1,m-1]</math>, <math>i \in [0,n-1]</math>, let <math>H'_{im+j} = H_{i+1} \cdot (H_i \cap K_j)</math>, and for integers <math>i \in [0,n]</math>, let |

<cmath> H'_{im} = H_i = H_{i+1} \cdot (H_i \cap K_0) = H_{i} \cdot (H_{i-1} \cap K_m), </cmath> | <cmath> H'_{im} = H_i = H_{i+1} \cdot (H_i \cap K_0) = H_{i} \cdot (H_{i-1} \cap K_m), </cmath> | ||

where these groups are defined. Similarly, for integers <math>i \in [1,n-1]</math>, <math>j\in [0,m-1]</math>, let <math>K'_{jn+i} = K_{j+1} \cdot (K_j \cap H_i)</math>, and for integers <math>j \in [0,m]</math>, define | where these groups are defined. Similarly, for integers <math>i \in [1,n-1]</math>, <math>j\in [0,m-1]</math>, let <math>K'_{jn+i} = K_{j+1} \cdot (K_j \cap H_i)</math>, and for integers <math>j \in [0,m]</math>, define |

## Revision as of 20:26, 21 January 2018

**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 and be composition series of a group . Then there exist equivalent composition series and such that is finer than and is finer than .

## Proof

Suppose and are the composition series in question. For integers , , let , and for integers , let where these groups are defined. Similarly, for integers , , let , and for integers , define where these groups are defined. Then by Zassenhaus's Lemma, and are composition series; they are evidently finer than and , respectively. Again by Zassenhaus's Lemma, the quotients and are equivalent, so series and are equivalent, as desired.