# Zassenhaus lemma

Hasse diagram of the Zassenhaus "butterfly" lemma - smaller subgroups are towards the top of the diagram

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.[1]

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

$B\triangleleft A$ and $D\triangleleft C$

are stable subgroups. Then,

$(A\cap C)B/(A\cap D)B$ is isomorphic to $(A\cap C)D/(B\cap C)D.$

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.

## Notes

1. ^ See Pierce, p. 27, exercise 1.