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The Schur–Zassenhaus theorem is a theorem in group theory which states that if $G$ is a finite group, and $N$ is a normal subgroup whose order is coprime to the order of the quotient group $G / N$ , then $G$ is a semidirect product of $N$ and $G / N$ .

An alternative statement of the theorem is that any normal Hall subgroup of a finite group $G$ has a complement in $G$ .

It is clear that if we do not impose the coprime condition, the theorem is not true: consider for example the cyclic group $C_{4}$ and its normal subgroup $C_{2}$ . Then if $C_{4}$ were a semidirect product of $C_{2}$ and $C_{4} / C_{2} ≅ C_{2}$ then $C_{4}$ would have to contain two elements of order 2, but it only contains one.

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 theory.

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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>.
+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>.
+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.
+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]].
+==References==
+*{{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}}
+*{{cite book | author=David S. Dummit & Richard M. Foote | title=Abstract Algebra | publisher=Wiley | year=2003 | isbn=978-0-471-43334-7}}
+{{DEFAULTSORT:Schur-Zassenhaus theorem}}
+[[Category:Theorems in group theory]]

Crosswordese: Difference between revisions

Revision as of 18:36, 24 January 2014

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