Zassenhaus lemma

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In mathematics, the butterfly lemma or Zassenhaus lemma is a technical result on the lattice of subgroups of a group.

Lemma (Butterfly lemma): Say <math>(G, \Omega)</math> is a group with operators and <math>A</math> and <math>C</math> are subgroups. Suppose <math>B\subset A</math> and <math>D\subset C</math> are stable 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.