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Finite Rational Matrix Groups

G. Nebe TWTH Aachen
W. Plesken Rheinisch-Westf Technology Hochschule
Available Formats:
Electronic ISBN: 978-1-4704-0135-1
Product Code: MEMO/116/556.E
List Price: $46.00 MAA Member Price:$41.40
AMS Member Price: $27.60 Click above image for expanded view Finite Rational Matrix Groups G. Nebe TWTH Aachen W. Plesken Rheinisch-Westf Technology Hochschule Available Formats:  Electronic ISBN: 978-1-4704-0135-1 Product Code: MEMO/116/556.E  List Price:$46.00 MAA Member Price: $41.40 AMS Member Price:$27.60
• Book Details

Memoirs of the American Mathematical Society
Volume: 1161995; 144 pp
MSC: 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 $p-1$ 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.

• 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 $16$
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 1161995; 144 pp
MSC: 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 $p-1$ 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.

• 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 $16$
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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