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Finite Rational Matrix Groups
G. Nebe and W. Plesken

Memoirs of the American Mathematical Society
1995; 144 pp; softcover
Volume: 116
ISBN-10: 0-8218-0343-3
ISBN-13: 978-0-8218-0343-1
List Price: US$46
Individual Members: US$27.60
Institutional Members: US$36.80
Order Code: MEMO/116/556
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The study of finite rational matrix groups reduces to the investigation of the maximal finite irreducible matrix groups and their natural lattices, which often turn out to have rather beautiful geometric and arithmetic properties. This book presents a full classification in dimensions up to 23 and with restrictions in dimensions and \(p+1\) and \(p-1\) for all prime numbers \(p\). Nonmaximal finite groups might act on several types of lattices and therefore embed into more than one maximal finite group. This gives rise to a simplicial complex interrelating the maximal finite groups and measuring the complexity of the dimension. Group theory, integral representation theory, arithmetic theory of quadratic forms and algorithmic methods are used.


Research mathematicians, researchers in group theory, number theory, discrete geometry, and coding theory.

Table of Contents

  • Finite rational matrix groups
  • Introduction
  • Notation, basic definitions, and constructions
  • Methods
  • Odd dimensions
  • Groups of type \(L_2(p)\) of degree \(p\pm 1\)
  • Dimensions \(2p\)
  • Dimension \(12\)
  • Dimension \(18\)
  • Dimension \(20\)
  • Appendix: The Gram matrices fixed by the primitive r.i.m.f. groups of degree \(n=12, 14, 15, 18, 20, 21\) and \(22\)
  • List of notations
  • References
  • Finite rational matrix groups of degree \(16\)
  • Introduction
  • Methods: Invariant quadratic forms and subgroups
  • The simplicial complexes \(M_8^{irr}(Q)\) and \(M_8^{irr,F}(Q)\)
  • Results in dimension \(16\)
  • Determination of the primitive r.i.m.f. groups of degree \(16\)
  • The simplicial complexes \(M_{16}^{irr}(Q)\) and \(M_{16}^{irr,F}(Q)\)
  • Appendix: The Gram matrices fixed by the primitive r.i.m.f. groups of degree \(16\)
  • References
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