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| 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.
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| 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]].
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| == Example ==
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| 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
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| : <math>\{[0]\} \times \{\operatorname{id}\} \; \triangleleft \; \mathbb{Z}/(2) \times \{\operatorname{id}\} \; \triangleleft \; \mathbb{Z}/(2) \times S_3,</math>
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| : <math>\{[0]\} \times \{\operatorname{id}\} \; \triangleleft \; \{[0]\} \times S_3 \; \triangleleft \; \mathbb{Z}/(2) \times S_3.</math> | |
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| <math>S_3</math> contains the normal subgroup <math>A_3</math>. Hence these have refinements
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| : <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>
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| with factor groups isomorphic to <math>(\mathbb{Z}/(2), A_3, \mathbb{Z}/(2))</math> and
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| : <math>\{[0]\} \times \{\operatorname{id}\} \; \triangleleft \; \{[0]\} \times A_3 \; \triangleleft \; \{[0]\} \times S_3 \; \triangleleft \; \mathbb{Z}/(2) \times S_3</math>
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| with factor groups isomorphic to <math>(A_3, \mathbb{Z}/(2), \mathbb{Z}/(2))</math>.
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| == References == | |
| *{{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}}
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| [[Category:Theorems in group theory]]
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| {{Abstract-algebra-stub}}
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