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Applications of the theory of orders to crystallographic groups

Part I

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Part of the Lecture Notes in Mathematics book series (LNM,volume 882)

Keywords

  • Finite Group
  • Point Group
  • Isomorphism Class
  • Finite Subgroup
  • Isomorphism Type

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References

  • [Abp 78] Abold, H., Plesken, W.: Ein Sylowsatz für endliche p-Untergruppen von GL(nL). Math. Ann. 232 (1978), 183–186.

    MathSciNet  CrossRef  MATH  Google Scholar 

  • [Bac 80] Backhouse, N.: Clifford theory and its application to the representation theory of crystallographic groups. To appear in [DrN 80].

    Google Scholar 

  • [BBNWZ 78] Brown, H., Bülow, R., Neubüser, J., Wondratschek, H., Zassenhaus H.: Crystallographic groups of four-dimensional space. Wiley, New York 1978.

    MATH  Google Scholar 

  • [Beh 62] Behr, H.: Über die endliche Definierbarkeit von Gruppen. Crelles Journal, 211 (1962), 116–122.

    MathSciNet  MATH  Google Scholar 

  • [Bie 10a] Bieberbach, L.: Über die Bewegungsgruppen des n-dimensionalen euclidischen Raumes mit einem endlichen Fundamentalbereich. Nachr. Königl. Ges. Wiss. Göttingen Math. Phys. K1 (1910) 75–84.

    Google Scholar 

  • [Bie 10b] Bieberbach, L. Über die Bewegungsgruppen der Euklidischen Räume. (Erste Abhandlung) Math. Ann., 70 (1910) 297–336.

    MathSciNet  CrossRef  MATH  Google Scholar 

  • [Bie 12] Bieberbach, L.: Über die Bewegungsgruppen der Euklidischen Räume. (Zweite Abhandlung) Die Gruppen mit einem endlichen Fundamentalbereich. Math. Ann. 72 (1912) 400–412.

    MathSciNet  CrossRef  MATH  Google Scholar 

  • [Bir 80] Birman, J. L.: Representation theory, selecton rules, and physical processes in crystals./Selection rules and symmetry breaking. To appear in [DrN 80]. Dress, A., Neubüser, J. (edit.): Proceedings of the interdisciplinary conference on crystallographic groups (Bielefeld 1978). To appear in MATCH/informal communications in mathematic chemistry (1980), (vol. 9 and 10).

    Google Scholar 

  • [BiS 28] Bieberbach, L., Schur, I.: Über Minkowskische Reduktionstheorie der positiven quadratischen Formen. Sitzungsber. der Preuss. Akad. der Wisensch. (1928), Physic.-Math. Klasse 510–535.

    Google Scholar 

  • [BNW 71] Bülow, R., Neubüser, J., Wondratschek, H.: On crystallography in higher dimensions. I. General definitions. II. Procedure of computations in R4. III. Resulsts in R4. Acta Crystallogr. A27 (1971), 517–535.

    CrossRef  Google Scholar 

  • [BNZ 72, 73] Brown, H., Neubüser, J., Zassenhaus, H.: On integral groups. I. The reducible case. Numer. Math. 19 (1972), 386–399. II. The irreducible case. Numer. Math. 20 (1972), 22–31. III. Normalizers. Math. Comput. 27 (1973), 167–182.

    MathSciNet  CrossRef  MATH  Google Scholar 

  • [BrM 77] Broderick, N., Maxwell G.: The crystallography of Coxeter groups. II. J. Algebra 44 (1977), 209–318.

    MathSciNet  CrossRef  MATH  Google Scholar 

  • [Bro 69] Brown, H.: An algorithm for the determination of space groups. Math. Comput. 23 (1969), 499–514.

    MathSciNet  CrossRef  MATH  Google Scholar 

  • [Bue 56] Buerger, M. J.: Elementary crystallography. An introduction to the fundamental geometrical features of crystals. Wiley, New York, 1956, Revised printing, Wiley, New York, 1963.

    MATH  Google Scholar 

  • [Bue 70] Buerger, M.J.: Contemporary crystallography. McGraw-Hill, New York, 1970.

    Google Scholar 

  • [Bül 67] Bülow, R. Eine Ableitung der Kristallklassen im R4 mit Hilfe gruppentheoretischer Programme. University of Kiel, 1967.

    Google Scholar 

  • [Bül 70] Bülow, R., Neubüser, J.: On some applications of group-theoretical programmes to the derivation of the crystal classes of R4. pp.131–135 in Leech, J., Ed.: Computational problems in abstract algebra (Proc. Conf. Oxford, 1967). Pergamon Press, Oxford, 1970.

    Google Scholar 

  • [Bül 73] Bülow, R.: Über Dadegruppen, in GL(5, ℤ). Dissertation, RWTH, Aachen, 1973.

    Google Scholar 

  • [Bur 66] Burckhardt, J.J.: Die Bewegungsgruppen der Kristallographie. 2nd ed., Birkhäuser, Basel, 1966.

    CrossRef  MATH  Google Scholar 

  • [CuR 62] Curtis, C.W., Reiner, I.: Representation theory of finite groups and associative algebras. Interscience, New York, 1962.

    MATH  Google Scholar 

  • [Cra 79] Cracknell, A.P., Davies, B.L., Miller, S.C., Love, W.F.: Kroncker Product Tables I–III. IFI/Plenum, New York, Washington, London 1979.

    Google Scholar 

  • [Dad 65] Dade, E.C.: The maximal finite groups of 4×4 matrices. III. J. Math. 9 (1965), 99–122.

    MathSciNet  MATH  Google Scholar 

  • [DrN 80] Dress, A., Neubüser, J. (edit.): Proceedings of the interdisciplinary conference on crystallographic groups (Bielefeld 1978). To appear in MATCH/informal communications in mathematic chemistry (1980), (vol. 9 and 10).

    Google Scholar 

  • [Fei 74] Feit, W.: On integral representations of finite groups. Proc. London Math. Soc. (3) 29 (1974), 633–683.

    MathSciNet  CrossRef  MATH  Google Scholar 

  • [Fel 63] Felsch, V., Neubüser, J.: Ein Programm zur Berechnung des Untergruppenverbandes einer endlichen Gruppe. Mitt. Rhein.-Westfäl. Inst. Instrum. Math. Bonn 2 (1963), 39–74.

    Google Scholar 

  • [FiNP 80] Finken, H., Neubüser, J., Plesken, W.: Space groups and groups of prime power-order. II. Classification of space groups by finite factor groups. Archiv d. Math. Vol. 35 (1980), 203–209.

    MathSciNet  CrossRef  MATH  Google Scholar 

  • [FNP 80] Felsch, W., Neubüser, J., Plesken, W.: Space groups and groups of prime-power order. IV. Counterexamples to the class-breadth conjecture. To appear Proceed. London Math. Soc.

    Google Scholar 

  • [Fro 11] Frobenius, F.G.: Über die unzerlegbaren diskreten Bewegungsgruppen. Sitzungsber. Preuss. Akad. Wiss. Berlin Phys. Math. Kl. (1911), 654–665. Gesammelte Abhandlungen III, Springer, Berlin, 1968, 507–518.

    Google Scholar 

  • [GrS 80] Grunewald, F., Segal, D.: Some general algorithms. I. Arithmetic groups. II. Nilpotent groups. To appear in Annals of Math.

    Google Scholar 

  • [Hen 69] Henry, N.F.M., Lonsdale, K., Eds.: International tables for X-ray crystallography. Vol. I. Symmetry groups. 3rd ed., The Kynoch Press, Birmingham, England, 1969.

    Google Scholar 

  • [Jan 80] Janner, A. G. M.: Symmetry of incommensurate crystal phases in the superspace group approach. To appear in [Drn 80]. Dress, A., Neubüser, J. (edit.): Proceedings of the interdisciplinary conference on crystallographic groups (Bielefeld 1978). To appear in MATCH/informal communications in mathematic chemistry (1980), (vol. 9 and 10).

    Google Scholar 

  • [Jar 80] Jarrat, J.D.: The decomposition of crystal families. To appear in Math. Proc. Cambridge Phil. Soc.

    Google Scholar 

  • [Jor 80] Jordan, C.: Mémoire sur l'équivalence des formes. J. Ecole Polytech. 48 (1880), 112–150. Oevres de C. Jordan, Vol. III, Gauthier-Villars, Paris, 1962, 421–460.

    Google Scholar 

  • [Köh 73] Köhler, K.-J.: Subperiodische kristallographische Bewegungsgruppen. Diplomarbeit, RWTH, Aachen, 1973.

    Google Scholar 

  • [Köh 80] Köhler, K.-J.: On the structure and the determination of n-dimensional partially periodic crystallographic groups. To appear in [Drn 80] Dress, A., Neubüser, J. (edit.): Proceedings of the interdisciplinary conference on crystallographic groups (Bielefeld 1978). To appear in MATCH/informal communications in mathematic chemistry (1980), (vol. 9 and 10).

    Google Scholar 

  • [LeN 80a, b] Leedham-Green, C.R., Newman, M. F.: Space groups and groups of prime-power order. I. Archiv d. Math. Vol. 35 (1980), 193–202. III. in preparation.

    MathSciNet  CrossRef  MATH  Google Scholar 

  • [Max 75] Maxwell, G.: The crystallography of Coxeter groups. J. Algebra 35 (1975), 159–177.

    MathSciNet  CrossRef  MATH  Google Scholar 

  • [Max 77] Maxwell, G.: Compact Euclidean space forms. J. Algebra 44 (1977), 191–195.

    MathSciNet  CrossRef  MATH  Google Scholar 

  • [Max 80] Maxwell, G.: Space groups of Coxeter type. To appear in [DrN 80] Dress, A., Neubüser, J. (edit.): Proceedings of the interdisciplinary conference on crystallographic groups (Bielefeld 1978). To appear in MATCH/informal communica

    Google Scholar 

  • [Mey 79] Meyer, J.: Les Puissances tensorielles de l'ideal augmentation d'un groupe fini et leurs extensions. These de doctorat, Grenoble 1979.

    Google Scholar 

  • [Min 87] Minkowski, H.: Über den arithmetischen Begriff der Äquivalenz und über die endlichen Gruppen linearer ganzzahliger Substitutionen. Crelles Journal 100 (1887), 449–458./ Zur Theorie der positiven quadratischen Formen, Crelles Journal 101 (1887), 196–202. Gesammelte Abhandlungen I, Teubner, Leipzig 1911, 201–218.

    MathSciNet  Google Scholar 

  • [Min 05] Minkowski, H.: Diskontinuitätsbereich für arithmetische Äquivalenz, J. Reine Angew. Math. 129 (1905), 220–274. Gesammelte Abhandlungen II, Teubner, Leipzig, 1911, 53–100.

    MathSciNet  Google Scholar 

  • [NPW 80] Neubüser, J., Plesken, W., Wondratschek, H.: An emendatory discursion on defining crystal system. To appear in [DrN 80] Dress, A., Neubüser, J. (edit.): Proceedings of the interdisciplinary conference on crystallographic groups (Bielefeld 1978). To appear in MATCH/informal communications in mathematic chemistry (1980), (vol. 9 and 10).

    Google Scholar 

  • [Ple 74] Beiträge zur Bestimmung der endlichen irreduziblen Untergruppen von GL(n, ℤ). Dissertation RWTH Aachen, 1974.

    Google Scholar 

  • [Ple 77] Plesken, W.: The Bravais group and the normalizer of a reducible finite subgroup of GL(n, ℤ). Comm. Algebra 5 (1977), 375–396.

    MathSciNet  CrossRef  MATH  Google Scholar 

  • [Ple 78] Plesken, W.: On reducible and decomposable representations of orders. Crelles Journal 297 (1978), 188–210.

    MathSciNet  MATH  Google Scholar 

  • [Ple 80a] Plesken, W.: Bravais groups in low dimensions. To appear in [DrN 80] Dress, A., Neubüser, J. (edit.): Proceedings of the interdisciplinary conference on crystallographic groups (Bielefeld 1978). To appear in MATCH/informal communications in mathematic chemistry (1980), (vol. 9 and 10).

    Google Scholar 

  • [Ple 80b] Plesken, W.: Gruppenringe über lokalen Dedekindbereichen. Habilitationsschrift Aachen 1980.

    Google Scholar 

  • [PlP 77,80] Plesken, W., Pohst, M.: On maximal finite irreducible subgroups of GL(n, ℤ). I. The five and seven dimensional case. II. The six dimensional case. III. The nine dimensional case. IV. Remarks on even dimensions with applications to n=8. V. The eight dimensional case and a complete description of dimensions less than ten. Math. Comput. 31 (1977), 536–577, and Math. Comput. 34, 149 (1980), 245–301.

    MathSciNet  MATH  Google Scholar 

  • [Rog 70] Roggenkamp, K. W.: Lattices over orders II. Springer Lecture Notes in Math. 142, Berlin, Heidelberg, New York 1970.

    Google Scholar 

  • [Rys 72a] Ryskov, S. S.: On maximal finite groups of integer (n×n)-matrices. Dokl. Akad. Nauk SSSR 204 (1972), 561–564. Sov. Math. Dokl. 13 (1972), 720–724.

    MathSciNet  Google Scholar 

  • [Rys 72b] Ryskov, S. S.: Maximal finite groups of integral n×n matrices and full groups of integral automorphisms of positive quadratic forms (Bravais models). Tr. Mat. Inst. Steklov 128 (1972), 183–211. Proc. Steklov Inst. Math. 128 (1972), 217–250.

    MathSciNet  Google Scholar 

  • [Sch 74] Schwarzenberger, R. L. E.: Crystallography in spaces of arbitrary dimension. Proc. Cambridge Philos. Soc. 76 (1974), 23–32.

    MathSciNet  CrossRef  MATH  Google Scholar 

  • [Sch 76] Schwarzenberger, R. L. E.: The use of directed graphs in the enumeration of orthogonal space groups. Acta Cryst. (1976), A 32, 556–559.

    CrossRef  Google Scholar 

  • [Sch 80a] Schwarzenberger, R. L. E.: N-dimensional crystallography. Research Notes in Mathematics 41. Pitman; San Francisco, London, Melbourne 1980.

    MATH  Google Scholar 

  • [Sch 80b] Graphical representation of n-dimensional space groups. To appear in Proc. London Math. Soc.

    Google Scholar 

  • [Sen 79] Senechal, M.: Color groups. Discrete Appl. Math. 1 (1979), 51–73.

    MathSciNet  CrossRef  MATH  Google Scholar 

  • [Sie 40] Siegel, C. L.: Einheiten quadratischer Formen. Abh. Math. Sem. Hamburg 13 (1940), 209–239.

    MathSciNet  CrossRef  MATH  Google Scholar 

  • [Sie 43] Siegel, C. L.: Discontinuous groups. Ann. Math. (2) 44 (1943), 674–689.

    MathSciNet  CrossRef  MATH  Google Scholar 

  • [Wol 67] Wolf, J. A.: Spaces of constant curvature. McGraw-Hill, New York, 1967.

    MATH  Google Scholar 

  • [Zas 38] Zassenhaus, H.: Neuer Beweis der Endlichkeit der Klassenzahl bei unimodularer Äquivalenz endlicher ganzzahliger Substitutiongruppen. Abh. Math. Sem. Hamburg 12 (1938), 276–288.

    MathSciNet  CrossRef  MATH  Google Scholar 

  • [Zas 48] Zassenhaus, H.: Über einen Algorithmus zur Bestimmung der Raumgruppen. Comment. Math. Helv. 21 (1948), 117–141.

    MathSciNet  CrossRef  MATH  Google Scholar 

  • [Zas 72] Zassenhaus, H.: On the units of orders. J. Alg. 20 (1972), 368–395.

    MathSciNet  CrossRef  MATH  Google Scholar 

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Plesken, W. (1981). Applications of the theory of orders to crystallographic groups. In: Roggenkamp, K.W. (eds) Integral Representations and Applications. Lecture Notes in Mathematics, vol 882. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0092487

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  • DOI: https://doi.org/10.1007/BFb0092487

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