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[[Image:Butterfly lemma.svg|thumb|300px|right|Hasse diagram of the Zassenhaus "butterfly" lemma - smaller subgroups are towards the top of the diagram]]
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In [[mathematics]], the '''butterfly lemma''' or '''Zassenhaus lemma''', named after [[Hans Zassenhaus]], is a technical result on the [[lattice of subgroups]] of a [[group (mathematics)|group]] or the [[lattice of submodules]] of a module, or more generally for any [[modular lattice]].<ref>See Pierce, p. 27, exercise 1.</ref>
 
'''Lemma:''' Suppose <math>(G, \Omega)</math> is a [[group with operators]] and <math>A</math> and <math>C</math> are [[subgroup]]s. Suppose
 
:<math>B\triangleleft A</math> and <math>D\triangleleft C</math>
 
are [[stable subgroup]]s. Then,
 
:<math>(A\cap C)B/(A\cap D)B</math> is [[isomorphism|isomorphic]] to <math>(A\cap C)D/(B\cap C)D.</math>
 
Zassenhaus proved this lemma specifically to give the smoothest proof of the [[Schreier refinement theorem]]. The 'butterfly' becomes apparent when trying to draw the [[Hasse diagram]] of the various groups involved.
 
==Notes==
<references/>
 
==References==
*{{citation|title=Associative algebras|first1=R. S.|last1=Pierce|publisher=Springer|pages=27|year=1982|isbn=0-387-90693-2}}.
*{{citation|title=An introduction to noncommutative noetherian rings|first1=K. R.|last1=Goodearl|first2=Robert B.|last2=Warfield|publisher=[[Cambridge University Press]]|year=1989|isbn=978-0-521-36925-1|pages=51, 62}}.
*{{citation|first=Serge|last=Lang|title=Algebra|pages=20–21|edition=Revised 3rd|series=Graduate Texts in Mathematics|publisher=[[Springer-Verlag]]|isbn=978-0-387-95385-4}}.
* Carl Clifton Faith, Nguyen Viet Dung, Barbara Osofsky (2009) ''Rings, Modules and Representations''. p.&nbsp;6. AMS Bookstore, ISBN 0-8218-4370-2
* [[Hans Zassenhaus]] (1934) "Zum Satz von Jordan-Hölder-Schreier", [[Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg]] 10:106&ndash;8.
* Hans Zassenhaus (1958) ''Theory of Groups'', second English edition, Lemma on Four Elements, p 74, [[Chelsea Publishing]].
 
==External links==
* Zassenhaus Lemma and proof at http://www.artofproblemsolving.com/Wiki/index.php/Zassenhaus%27s_Lemma
 
{{DEFAULTSORT:Zassenhaus Lemma}}
[[Category:Group theory]]
[[Category:Lemmas]]
[[Category:Isomorphism theorems]]

Latest revision as of 13:39, 24 November 2014

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