In mathematics, the butterfly lemma or Zassenhaus lemma, named after Hans Julius Zassenhaus, is a technical result on the lattice of subgroups of a group or the lattice of submodules of a module, or more generally for any modular lattice.[1]
Lemma: Suppose \( (G, \Omega) \) is a group with operators and A and C are subgroups. Suppose
\( B\triangleleft A and D\triangleleft C \)
are stable subgroups. Then,
\( (A\cap C)B/(A\cap D)B is isomorphic to (A\cap C)D/(B\cap C)D. \)
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.
Hasse diagram of the Zassenhaus "butterfly" lemma - smaller subgroups are towards the top of the diagram(*)
Algebra Lecture 9.3: Zassenhaus Butterfly Lemma, Part 1
Algebra Lecture 9.4: Zassenhaus Butterfly Lemma, Part 2
Notes
^ See Pierce, p. 27, exercise 1.
References
Pierce, R. S., Associative algebras, Springer, pp. 27, ISBN 0387906932.
Goodearl, K. R.; Warfield, Robert B. (1989), An introduction to noncommutative noetherian rings, Cambridge University Press, pp. 51, 62, ISBN 9780521369251.
Lang, Serge, Algebra, Graduate Texts in Mathematics (Revised 3rd ed.), Springer-Verlag, pp. 20–21, ISBN 9780387953854.
Carl Clifton Faith, Nguyen Viet Dung, Barbara Osofsky. Rings, Modules and Representations. p. 6. AMS Bookstore, 2009. ISBN 0821843702
External links
Zassenhaus Lemma and proof at http://www.artofproblemsolving.com/Wiki/index.php/Zassenhaus%27s_Lemma
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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