On integral groups. III. Normalizers
HTML articles powered by AMS MathViewer
- by H. Brown, J. Neubüser and H. Zassenhaus PDF
- Math. Comp. 27 (1973), 167-182 Request permission
Abstract:
Methods for determining a generating set for the normalizer of a finite group of $n \times n$ integral matrices, i.e., an n-dimensional crystallographic point group, are discussed. Necessary and sufficient conditions for the finiteness of such a normalizer are derived, and several examples of the application of the methods to cases when the normalizer is infinite are presented.References
- A. Adrian Albert, Structure of algebras, American Mathematical Society Colloquium Publications, Vol. XXIV, American Mathematical Society, Providence, R.I., 1961. Revised printing. MR 0123587
- Harold Brown, An algorithm for the determination of space groups, Math. Comp. 23 (1969), 499–514. MR 246975, DOI 10.1090/S0025-5718-1969-0246975-6
- H. Brown, J. Neubüser, and H. Zassenhaus, On integral groups. I. The reducible case, Numer. Math. 19 (1972), 386–399. MR 318334, DOI 10.1007/BF01404921 H. Brown, J. Neubüser & H. Zassenhaus, "On integral groups. II," Numer. Math., v. 19, 1972. pp. 297-299.
- Harold Brown, An application of Zassenhaus’ unit theorem, Acta Arith. 20 (1972), 297–298. MR 304424, DOI 10.4064/aa-20-3-297-298 R. Bülow, Schreier program (Unpublished.) R. Bülow & J. Neubüser, "On some applications of group-theoretical programmes to the derivation of the crystal classes of ${R_4}$," Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), Pergamon Press, Oxford, 1970, pp. 131-135. MR 41 #5504. R. Bülow, J. Neubüser & H. Wondratschek, "On crystallography in higher dimensions. II," Acta Cryst., v. A27, 1971, pp. 520-523. C. Chevalley, L’Arithmétique dans les Algébres des Matrices, Actualités Sci. Indust., no. 323, Hermann, Paris, 1936.
- Irving Reiner, A survey of integral representation theory, Bull. Amer. Math. Soc. 76 (1970), 159–227. MR 254092, DOI 10.1090/S0002-9904-1970-12441-7
- Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0144979
- E. C. Dade, The maximal finite groups of $4\times 4$ integral matrices, Illinois J. Math. 9 (1965), 99–122. MR 170958
- Helmut Hasse, Zahlentheorie, Akademie-Verlag, Berlin, 1963 (German). Zweite erweiterte Auflage. MR 0153659
- Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1966. MR 0207802
- Carl Ludwig Siegel, Discontinuous groups, Ann. of Math. (2) 44 (1943), 674–689. MR 9959, DOI 10.2307/1969104
- G. L. Watson, Integral quadratic forms, Cambridge Tracts in Mathematics and Mathematical Physics, No. 51, Cambridge University Press, New York, 1960. MR 0118704
- Hans Zassenhaus, Über einen Algorithmus zur Bestimmung der Raumgruppen, Comment. Math. Helv. 21 (1948), 117–141 (German). MR 24424, DOI 10.1007/BF02568029
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Math. Comp. 27 (1973), 167-182
- MSC: Primary 20H15
- DOI: https://doi.org/10.1090/S0025-5718-1973-0333025-7
- MathSciNet review: 0333025