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In mathematics, the butterfly lemma or Zassenhaus lemma is a technical result on the lattice of subgroups of a group.
First, a definition. A group, <math>G<math>, is an <math>\Omega<math>-group if and only if there exists a set map
- <math>\Omega\rightarrow\mathrm{End}_{\mathbf{Grp}}(G)<math>,
where <math>\mathbf{Grp}<math> is the category of groups and <math>\mathrm{End}_{\mathbf{Grp}}(G)<math> is the set of group endomorphisms of <math>G<math>.
Lemma (Butterfly lemma): Say <math>G<math> is an <math>\Omega<math>-group and <math>A<math> and <math>C<math> are subgroups. Suppose
<math>B\subset A<math> and <math>D\subset C<math> are <math>\Omega<math>-subgroups. Then,
- <math>(A\cap C)B/(A\cap D)B<math> is isomorphic to <math>(C\cap A)D/(C\cap B)D.<math>
Hans Julius 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.
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