Butterfly lemma

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.

Lemma: Suppose ${\displaystyle (G,\Omega )}$ is a group with operators and ${\displaystyle A}$ and ${\displaystyle C}$ are subgroups. Suppose

are stable subgroups. Then,

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