Zassenhaus lemma

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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 (G,Ω) is a group with operators and A and C 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.