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.
Lemma: Suppose is a group with operators and and are subgroups. Suppose
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.