View source for Schreier refinement theorem

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 groups that sends each [[factor group]] to an 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]]. There are subnormal series

: <math>\{[0]\} \times \{\operatorname{id}\} \; \triangleleft \; \mathbb{Z}/(2) \times \{\operatorname{id}\} \; \triangleleft \; \mathbb{Z}/(2) \times S_3,</math>
: <math>\{[0]\} \times \{\operatorname{id}\} \; \triangleleft \; \{[0]\} \times S_3 \; \triangleleft \; \mathbb{Z}/(2) \times S_3.</math>

<math>S_3</math> contains the normal subgroup <math>A_3</math>. Hence these have refinements

: <math>\{[0]\} \times \{\operatorname{id}\} \; \triangleleft \; \mathbb{Z}/(2) \times \{\operatorname{id}\} \; \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 \{\operatorname{id}\} \; \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}}

*{{cite book | author=Rotman, Joseph  | title=An introduction to the theory of groups | location=New York | publisher=Springer-Verlag | year=1994 | isbn=0-387-94285-8}}
[[Category:Theorems in group theory]]

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