Electronic ISBN:  9781470401351 
Product Code:  MEMO/116/556.E 
List Price:  $46.00 
MAA Member Price:  $41.40 
AMS Member Price:  $27.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 116; 1995; 144 ppMSC: Primary 20; 11;
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.
ReadershipResearch mathematicians, researchers in group theory, number theory, discrete geometry, and coding theory.

Table of Contents

Chapters

Finite rational matrix groups

I. Introduction

II. Notation, basic definitions, and constructions

III. Methods

IV. Odd dimensions

V. Groups of type $L_2(p)$ of degree $p \pm 1$

VI. Dimensions $2p$

VII. Dimension 12

VIII. Dimension 18

IX. 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 $22$

Finite rational matrix groups of degree $16$

I. Introduction

II. Methods: Invariant quadratic forms and subgroups

III. The simplicial complexes $M^{irr}_8(\mathbb {Q})$ and $M^{irr,F}_8(\mathbb {Q})$

IV. Results in dimension 16

V. Determination of the primitive r.i.m.f. groups of degree 16

VI. The simplicial complexes $M^{irr}_{16}(\mathbb {Q})$ and $M^{irr,F}_{16}(\mathbb {Q})$

Appendix. The Gram matrices fixed by the primitive r.i.m.f. groups of degree $16$


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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 \(p1\) 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.
Research mathematicians, researchers in group theory, number theory, discrete geometry, and coding theory.

Chapters

Finite rational matrix groups

I. Introduction

II. Notation, basic definitions, and constructions

III. Methods

IV. Odd dimensions

V. Groups of type $L_2(p)$ of degree $p \pm 1$

VI. Dimensions $2p$

VII. Dimension 12

VIII. Dimension 18

IX. 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 $22$

Finite rational matrix groups of degree $16$

I. Introduction

II. Methods: Invariant quadratic forms and subgroups

III. The simplicial complexes $M^{irr}_8(\mathbb {Q})$ and $M^{irr,F}_8(\mathbb {Q})$

IV. Results in dimension 16

V. Determination of the primitive r.i.m.f. groups of degree 16

VI. The simplicial complexes $M^{irr}_{16}(\mathbb {Q})$ and $M^{irr,F}_{16}(\mathbb {Q})$

Appendix. The Gram matrices fixed by the primitive r.i.m.f. groups of degree $16$