Editing Schreier refinement theorem

{{Short description|Statement in group theory}}
In [[mathematics]], the '''Schreier refinement theorem''' of [[group theory]] states that any two [[subnormal series]] of [[subgroup]]s of a given group have equivalent refinements, where two series are equivalent if there is a [[bijection]] between their [[factor group]]s that sends each factor group to an [[group isomorphism|isomorphic]] one.

The theorem is named after the [[Austria]]n [[mathematician]] [[Otto Schreier]] who proved it in 1928. It provides an elegant proof of the [[Jordan–Hölder theorem]]. It is often proved using the [[Zassenhaus lemma]]. {{harvtxt|Baumslag|2006}} gives a short proof by intersecting the terms in one subnormal series with those in the other series.

== Example ==

Consider <math>\mathbb{Z}_2 \times S_3</math>, where <math>S_3</math> is the [[symmetric group of degree 3]].  The [[alternating group]] <math>A_3</math> is a normal subgroup of <math>S_3</math>, so we have the two subnormal series
<div style="text-align: center;">
: <math>\{0\} \times \{(1)\} \; \triangleleft \; \mathbb{Z}_2 \times \{(1)\} \; \triangleleft \; \mathbb{Z}_2 \times S_3,</math>
: <math>\{0\} \times \{(1)\} \; \triangleleft \; \{0\} \times A_3 \; \triangleleft \; \mathbb{Z}_2 \times S_3,</math>
</div>
with respective factor groups <math>(\mathbb{Z}_2,S_3)</math> and <math>(A_3,\mathbb{Z}_2\times\mathbb{Z}_2)</math>.<br>
The two subnormal series are not equivalent, but they have equivalent refinements:

: <math>\{0\} \times \{(1)\} \; \triangleleft \; \mathbb{Z}_2 \times \{(1)\} \; \triangleleft \; \mathbb{Z}_2 \times A_3 \; \triangleleft \; \mathbb{Z}_2 \times S_3</math>

with factor groups isomorphic to <math>(\mathbb{Z}_2, A_3, \mathbb{Z}_2)</math> and

: <math>\{0\} \times \{(1)\} \; \triangleleft \; \{0\} \times A_3 \; \triangleleft \; \{0\} \times S_3 \; \triangleleft \; \mathbb{Z}_2 \times S_3</math>

with factor groups isomorphic to <math>(A_3, \mathbb{Z}_2, \mathbb{Z}_2)</math>.

== References ==

* {{citation|author=Baumslag|first=Benjamin|title=A simple way of proving the Jordan-Hölder-Schreier theorem|journal=American Mathematical Monthly|volume=113|year=2006|issue=10|
pages=933–935|doi=10.2307/27642092|jstor=27642092}}

[[Category:Theorems in group theory]]


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