|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| 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>.
| | She is recognized by the title of Myrtle Shryock. Her family members lives in Minnesota. Hiring has been my profession for some time but I've already utilized for an additional 1. It's not a typical factor but what she likes performing is foundation jumping and now she is attempting to earn money with it.<br><br>Also visit my blog; [http://www.animecontent.com/blog/349119 animecontent.com] |
| | |
| 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]]
| |
Latest revision as of 20:51, 18 November 2014
She is recognized by the title of Myrtle Shryock. Her family members lives in Minnesota. Hiring has been my profession for some time but I've already utilized for an additional 1. It's not a typical factor but what she likes performing is foundation jumping and now she is attempting to earn money with it.
Also visit my blog; animecontent.com