Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Irreducible finite integral matrix groups of degree $ 8$ and $ 10$


Author: Bernd Souvignier
Journal: Math. Comp. 63 (1994), 335-350
MSC: Primary 20H15; Secondary 11E12, 20C10, 20C40
DOI: https://doi.org/10.1090/S0025-5718-1994-1213836-X
MathSciNet review: 1213836
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The lattices of eight- and ten-dimensional Euclidean space with irreducible automorphism group or, equivalently, the conjugacy classes of these groups in $ \mathrm{GL}_n(\mathbb{Z})$ for $ n = 8,10$, are classified in this paper. The number of types is 52 in the case $ n = 8$, and 47 in the case $ n = 10$. As a consequence of this classification one has 26, resp. 46, conjugacy classes of maximal finite irreducible subgroups of $ \mathrm{GL}_8(\mathbb{Z})$, resp. $ \mathrm{GL}_{10}(\mathbb{Z})$. In particular, each such group is absolutely irreducible, and therefore each of the maximal finite groups of degree 8 turns up in earlier lists of classifications.


References [Enhancements On Off] (What's this?)

  • [1] H. Brown, R. Bülow, J. Neubüser, H. Wondratschek, and H. Zassenhaus, Crystallographic groups of four-dimensional space, Wiley-Interscience, New York, 1978.
  • [2] H. F. Blichfeldt, Finite collineation groups, Univ. of Chicago Press, Chicago, IL, 1917.
  • [3] S. I. Borewicz and I. R. Šafarevič, Zahlentheorie, Birkhäuser, Basel, 1966.
  • [4] J. Cannon, A general purpose group theory program, Lecture Notes in Math., vol. 372, Springer-Verlag, Berlin and New York, 1973. MR 0354823 (50:7300)
  • [5] M. Eichler, Über die Klassenzahl total definiter Quaternionenalgebren, Math. Z. 43 (1938), 102-109. MR 1545717
  • [6] M. Schönert (ed.), GAP: groups, algorithms and programming. Manual (version 3.1), Lehrstuhl D für Mathematik, RWTH Aachen, 1992.
  • [7] H. Hasse, Über die Klassenzahl abelscher Zahlkörper, Akademie-Verlag, Berlin, 1952. MR 0049239 (14:141a)
  • [8] D. Holt and W. Plesken, Perfect groups, Oxford Univ. Press, Oxford, 1989. MR 1025760 (91c:20029)
  • [9] H. Lang, Über eine Gattung elementar-arithmetischer Klasseninvarianten reell-quadratischer Zahlkörper, J. Reine Angew. Math. 233 (1968), 123-175. MR 0238804 (39:168)
  • [10] W. Plesken, Some applications of representation theory, Progr. Math., vol. 95, Birkhäuser, Basel, 1991, pp. 477-496. MR 1112176 (92k:20019)
  • [11] W. Plesken and W. Hanrath, The lattices of six-dimensional Euclidean space, Math. Comp. 43 (1984), 573-587. MR 758205 (85m:11036)
  • [12] W. Plesken and M. Pohst, On maximal finite irreducible subgroups of $ \mathrm{GL}(n,\mathbb{Z})$. I and II, Math. Comp. 31 (1977), 536-573. MR 0444789 (56:3137a)
  • [13] -, On maximal finite irreducible subgroups of $ \mathrm{GL}(n,\mathbb{Z})$. III, IV and V, Math. Comp. 34 (1980), 245-301. MR 551303 (81b:20012a)
  • [14] -, Constructing integral lattices with prescribed minimum. I, Math. Comp. 45 (1985), 209-221. MR 790654 (87e:11077)
  • [15] I. Reiner, Maximal orders, Academic Press, London, 1975. MR 1972204 (2004c:16026)
  • [16] C. C. Sims, Computation methods in the study of permutation groups, Computational Problems in Abstract Algebra (J. Leech, ed.), Pergamon Press, Oxford, 1970. MR 0257203 (41:1856)
  • [17] B. Souvignier, Irreduzible Bravaisgruppen in $ \mathrm{GL}_8(\mathbb{Z})$, Diplomarbeit RWTH Aachen, 1991.
  • [18] L. C. Washington, Introduction to cyclotomic fields, Springer, New York, 1982. MR 718674 (85g:11001)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 20H15, 11E12, 20C10, 20C40

Retrieve articles in all journals with MSC: 20H15, 11E12, 20C10, 20C40


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1994-1213836-X
Keywords: Integral matrix groups, Bravais groups, integral lattices in Euclidean space
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society