In mathematics, the butterfly lemma or Zassenhaus lemma, named after Hans 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 is a group with subgroups and . Suppose and are normal subgroups. Then there is an isomorphism of quotient groups:
This can be generalized to the case of a group with operators with stable subgroups and , the above statement being the case of acting on itself by conjugation.
Zassenhaus proved this lemma specifically to give the most direct proof of the Schreier refinement theorem. The 'butterfly' becomes apparent when trying to draw the Hasse diagram of the various groups involved.
Zassenhaus' lemma for groups can be derived from a more general result known as Goursat's theorem stated in a Goursat variety (of which groups are an instance); however the group-specific modular law also needs to be used in the derivation.[2]
^Pierce, R.S. (1982). Associative algebras. Springer. p. 27, exercise 1. ISBN 0-387-90693-2.
^J. Lambek (1996). "The Butterfly and the Serpent". In Aldo Ursini; Paulo Agliano (eds.). Logic and Algebra. CRC Press. pp. 161–180. ISBN 978-0-8247-9606-8.
In mathematics, the butterfly lemma or Zassenhauslemma, named after Hans Zassenhaus, is a technical result on the lattice of subgroups of a group or the...
(1912–1991), German mathematician Zassenhaus algorithm Zassenhaus group Zassenhauslemma Hiltgunt Zassenhaus (1916–2004), German philologist who aided Scandinavian...
theory. In 1959 Zassenhaus began teaching at University of Notre Dame and became director of its computing center in 1964. Zassenhaus was a Mershon visiting...
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well-known in German-speaking countries for a long time. According to Hans Zassenhaus: O. Schreier's and Artin's ingenious characterization of formally real...
referred to simply as semidirect products. For finite groups, the Schur–Zassenhaus theorem provides a sufficient condition for the existence of a decomposition...
f_{r}} each of degree d. We first describe an algorithm by Cantor and Zassenhaus (1981) and then a variant that has a slightly better complexity. Both...
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g(x)} can be reconstructed from its image mod m {\displaystyle m} . The Zassenhaus algorithm proceeds as follows. First, choose a prime number p {\displaystyle...
factorization via e.g., over finite fields, Berlekamp's algorithm or Cantor–Zassenhaus algorithm. Greatest common divisor via e.g. Euclidean algorithm Gaussian...
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This decomposition is also a consequence (particular case) of the Schur–Zassenhaus theorem. In terms of permutations the two group elements of G / A3 are...