Butterfly Lemma


Given two normal subgroups G_1 and G_2 of a group, and two normal subgroups H_1 and H_2 of G_1 and G_2 respectively,

 H_1(G_1 intersection H_2)    is normal in    H_1(G_1 intersection G_2)
 (H_1 intersection G_2)H_2    is normal in    (G_1 intersection G_2)H_2,

and one has an isomorphism of quotient groups

 H_1(G_1 intersection G_2)/H_1(G_1 intersection H_2)=(G_1 intersection G_2)H_2/(H_1 intersection G_2)H_2

(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.

See also

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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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.", 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.

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Butterfly Lemma

Cite this as:

Barile, Margherita. "Butterfly Lemma." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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