Memoirs of the American Mathematical Society 1995; 144 pp; softcover Volume: 116 ISBN10: 0821803433 ISBN13: 9780821803431 List Price: US$43 Individual Members: US$25.80 Institutional Members: US$34.40 Order Code: MEMO/116/556
 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 \(p1\) 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. Readership 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
