# Finite Rational Matrix Groups

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*G. Nebe; W. Plesken*

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.

#### Table of Contents

# Table of Contents

## Finite Rational Matrix Groups

- Table of Contents vii8 free
- Finite rational matrix groups 110 free
- I. Introduction 110
- II. Notation, basic definitions, and constructions 413
- III. Methods 1322
- IV. Odd dimensions 1928
- V. Groups of type L[sub(2)](p) of degree p±l 2231
- VI. Dimensions 2[sub(p)] 3342
- VII. Dimension 12 3645
- VIII. Dimension 18 4453
- IX. Dimension 20 4958
- Appendix: The Gram matrices fixed by the primitive r.i.m.f groups of degree n = 12,14,15,18, 20,21 and 22 6473
- List of notations 7079
- References 7281

- Finite rational matrix groups of degree 16 7483
- I. Introduction 7483
- II. Methods: Invariant quadratic forms and subgroups 7685
- III. The simplicial complexes M[sup(irr)][sub(8)](Q) and M[sup(irr,F)][sub(8)](Q) 8493
- IV. Results in dimension 16 8695
- V. Determination of the primitive r.i.m.f. groups of degree 16 9099
- VI. The simplicial complexes M[sup(irr}][sub(16)](Q) and M[sup(irr,F)][sub(16)](Q) 114123
- Appendix: The Gram matrices fixed by the primitive r.i.m.f groups of degree 16 141150
- References 144153