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| The '''Schur–Zassenhaus theorem''' is a [[theorem]] in [[group theory]] which states that if <math>G</math> is a finite [[group (mathematics)|group]], and <math>N</math> is a [[normal subgroup]] whose [[order (group theory)|order]] is [[coprime]] to the order of the [[quotient group]] <math>G/N</math>, then <math>G</math> is a [[semidirect product]] of <math>N</math> and <math>G/N</math>. | | The title of [http://nxnn.info/user/G8498 over the counter std test] author is Figures. My working day job is a meter reader. To play baseball is the pastime he will by no means quit doing. Years ago we moved to Puerto Rico and my family members loves it. |
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| An alternative statement of the theorem is that any normal [[Hall subgroup]] of a finite group <math>G</math> has a [[complement (group theory)|complement]] in <math>G</math>.
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| It is clear that if we do not impose the coprime condition, the theorem is not [[truth|true]]: consider for example the [[cyclic group]] <math>C_4</math> and its normal subgroup <math>C_2</math>. Then if <math>C_4</math> were a semidirect product of <math>C_2</math> and <math>C_4 / C_2 \cong C_2</math> then <math>C_4</math> would have to contain two [[element (mathematics)|element]]s of order 2, but it only contains one.
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| The Schur–Zassenhaus theorem at least partially answers the question: "In a [[composition series]], how can we classify groups with a certain set of composition factors?" The other part, which is where the composition factors do not have coprime orders, is tackled in [[extension problem|extension theory]].
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| ==References==
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| *{{cite book | author=Rotman, Joseph J. | title=An Introduction to the Theory of Groups | location=New York | publisher=Springer–Verlag | year=1995 | isbn=978-0-387-94285-8}}
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| *{{cite book | author=David S. Dummit & Richard M. Foote | title=Abstract Algebra | publisher=Wiley | year=2003 | isbn=978-0-471-43334-7}}
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| {{DEFAULTSORT:Schur-Zassenhaus theorem}}
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| [[Category:Theorems in group theory]]
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Revision as of 17:54, 25 February 2014
The title of over the counter std test author is Figures. My working day job is a meter reader. To play baseball is the pastime he will by no means quit doing. Years ago we moved to Puerto Rico and my family members loves it.