Lattice Isomorphisms and Saturated Formations of Finite Soluble Groups

  • Roland Schmidt
Part of the Lecture Notes in Mathematics book series (LNM, volume 372)

Abstract

Let G and be (finite) groups and let α be an isomorphism from the subgroup lattice L(G) of G onto L(). Then in general G and need not he isomorphic. So it is natural to ask which group theoretic properties groups with isomorphic subgroup lattices have in common. We call a class K of groups invariant under lattice isomorphisms (short: lattice-invariant) if GK implies K for every group with L() isomorphic to L(G). Then our question raised above is what classes of groups are lattice-invariant.

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References

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    Wolfgang Gaschütz, “Zur Theorie der endlichen auflösbaren Gruppen”, Math. Z. 80 (1963), 300–305. MR31#3505.MathSciNetCrossRefMATHGoogle Scholar
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    B. Huppert, Endliche Gruppen I (Die Grundlehren der mathematischen Wissenschaften, Band 134. Springer-Verlag, Berlin, Heidelberg, New York, 1967). MR37#302.Google Scholar
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    Roland Schmidt, “Verbandsisomorphismen endlicher auflösbarer Gruppen”, Arch. Math. 23 (1972), 449–458.CrossRefMATHGoogle Scholar
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    Michio Suzuki, Structure of a group and the structure of its lattice of subgroups (Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Heft 10. Springer-Verlag, Berlin, Göttingen, Heidelberg, 1956). MR18,715.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1974

Authors and Affiliations

  • Roland Schmidt
    • 1
  1. 1.Mathematisches Seminar der Christian-Albrechts-Universität23 Kiel(West) Germany

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