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# Butterfly Lemma

Given two normal subgroups and of a group, and two normal subgroups and of and respectively,

 (1)
 (2)

and one has an isomorphism of quotient groups

 (3)

(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.

The butterfly lemma can be used to prove the equivalence of composition series in Jordan-Hölder theorem.

Butterfly Catastrophe, Butterfly Curve, Butterfly Effect, Butterfly Function, Butterfly Graph, Butterfly Polyiamond, Butterfly Theorem, Composition Series, Hasse Diagram, Jordan-Hölder Theorem

This entry contributed by Margherita Barile

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## References

Lang, S. Algebra, 3rd rev. ed. New York: Springer-Verlag, 2002.Rotman, J. J. An Introduction to the Theory of Groups, 2nd rev. ed. Newton, MA: Allyn and Bacon, pp. 77-78, 1984.Smith, T. L. "Composition Series and Solvable Groups." http://math.uc.edu/~tsmith/Math610/compseries.pdf.Zassenhaus, H. J. "Zum Satz von Jordan-Hölder-Schreier." Abh. Math. Semin. Hamb. Univ. 10, 106-108, 1934.Zassenhaus, H. J. The Theory of Groups, 2nd ed. New York: Chelsea, pp. 38-39, 1974.

Butterfly Lemma

## Cite this as:

Barile, Margherita. "Butterfly Lemma." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ButterflyLemma.html