# Difference between revisions of "Schreier's Theorem"

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## Latest revision as of 12:13, 9 April 2019

**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.