Zassenhaus theorem butterfly lemma proof
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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 or the lattice of submodules of a module, or more generally for any modular lattice .[1]
Lemma: Suppose is a group with operators and and are subgroups . Suppose
and
are stable subgroups . Then,
is isomorphic to
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.
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]
Notes
1. ^ See Pierce, p. 27, exercise 1.
2. ^ J. Lambek (1996). "The Butterfly and the Serpent". In Aldo Ursini, Paulo Agliano. Logic and Algebra . CRC Press. pp. 161–180.
ISBN 978-0-8247-9606-8 .
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 operators and and are subgroups . Suppose
and
are stable subgroups . Then,
is isomorphic to
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.
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]
Notes
1. ^ See Pierce, p. 27, exercise 1.
2. ^ J. Lambek (1996). "The Butterfly and the Serpent". In Aldo Ursini, Paulo Agliano. Logic and Algebra . CRC Press. pp. 161–180.
ISBN 978-0-8247-9606-8 .
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