(Zassenhaus 1934). This lemma was named by Serge Lang (2002, pp. 20-21) based on the shape of the diagram above, which Lang derived from
Zassenhaus's original publication.
The butterfly lemma visualizes the inclusion between subgroups. In particular, whenever two groups are connected by a segment to a point lying right above, this point represents
their product, and whenever the point lies right below, it represents their intersection.
This diagram is part of the Hasse diagram of the
partially ordered set of subgroups of the
given group. The quotient groups formed along the
three central vertical lines are all isomorphic.
and 
of a group, and two normal
subgroups 
and 
of 
and 
respectively,
