Irreducible finite integral matrix groups of degree 8 and 10

Souvignier, Bernd

doi:10.1090/S0025-5718-1994-1213836-X

Irreducible finite integral matrix groups of degree $8$ and $10$
HTML articles powered by AMS MathViewer

by Bernd Souvignier PDF

Math. Comp. 63 (1994), 335-350 Request permission

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

Crystallographic groups of four-dimensional space

Finite collineation groups

Zahlentheorie

John Cannon, A general purpose group theory program, Proceedings of the Second International Conference on the Theory of Groups (Australian Nat. Univ., Canberra, 1973) Lecture Notes in Math., Vol. 372, Springer, Berlin, 1974, pp. 204–217. MR 0354823
M. Eichler, Über die Idealklassenzahl total definiter Quaternionenalgebren, Math. Z. 43 (1938), no. 1, 102–109 (German). MR 1545717, DOI 10.1007/BF01181088

GAP

groups, algorithms and programming. Manual

version 3.1

Helmut Hasse, Über die Klassenzahl abelscher Zahlkörper, Akademie-Verlag, Berlin, 1952 (German). MR 0049239
Derek F. Holt and W. Plesken, Perfect groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1989. With an appendix by W. Hanrath; Oxford Science Publications. MR 1025760
Heinrich Lang, Über eine Gattung elementar-arithmetischer Klasseninvarianten reell-quadratischer Zahlkörper, J. Reine Angew. Math. 233 (1968), 123–175 (German). MR 238804, DOI 10.1515/crll.1968.233.123
W. Plesken, Some applications of representation theory, Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991) Progr. Math., vol. 95, Birkhäuser, Basel, 1991, pp. 477–496. MR 1112176, DOI 10.1007/978-3-0348-8658-1_{2}2
W. Plesken and W. Hanrath, The lattices of six-dimensional Euclidean space, Math. Comp. 43 (1984), no. 168, 573–587. MR 758205, DOI 10.1090/S0025-5718-1984-0758205-5
Wilhelm Plesken and Michael Pohst, On maximal finite irreducible subgroups of $\textrm {GL}(n, \textbf {Z})$. I. The five and seven dimensional cases, Math. Comp. 31 (1977), no. 138, 536–551. MR 444789, DOI 10.1090/S0025-5718-1977-0444789-X
Wilhelm Plesken and Michael Pohst, On maximal finite irreducible subgroups of $\textrm {GL}(n,\,\textbf {Z})$. III. The nine-dimensional case, Math. Comp. 34 (1980), no. 149, 245–258. MR 551303, DOI 10.1090/S0025-5718-1980-0551303-7
W. Plesken and M. Pohst, Constructing integral lattices with prescribed minimum. I, Math. Comp. 45 (1985), no. 171, 209–221, S5–S16. MR 790654, DOI 10.1090/S0025-5718-1985-0790654-2
I. Reiner, Maximal orders, London Mathematical Society Monographs. New Series, vol. 28, The Clarendon Press, Oxford University Press, Oxford, 2003. Corrected reprint of the 1975 original; With a foreword by M. J. Taylor. MR 1972204
Charles C. Sims, Computational methods in the study of permutation groups, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) Pergamon, Oxford, 1970, pp. 169–183. MR 0257203

Irreduzible Bravaisgruppen in

Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674, DOI 10.1007/978-1-4684-0133-2