Generators and Relations for Discrete Groups - Classic Reference

In this review of Generators and Relations for Discrete Groups the bottom line is clear: this is a specialist reference for mathematicians and advanced students who need a compact catalogue of abstract definitions and small-order group data. The reviewer finds the book most valuable for its curated tables and focused scope rather than as an introductory textbook. It is concise, historically grounded, and best used as a supplementary reference when working with permutation groups, congruent transformation groups, or reflection-generated groups in Euclidean space.

Key Features

• Curated tables: The book includes tables covering low-order groups and specific classes that save time when searching for concrete examples and presentations.
• Coverage of permutation groups: It addresses permutation groups of small degree and points readers to classical tables for exhaustive listings.
• Reflection groups: Groups generated by reflections in real Euclidean space are treated, which aids work on symmetry and geometric group theory.
• Linear fractional and symmetric groups: The text presents definitions and material on linear fractional, symmetric, and alternating groups useful for algebraic classification tasks.
• Finite and infinite congruent transformation groups: The reviewer notes useful entries on both finite and infinite groups arising from congruent transformations.
• Concise editorial approach: The authors restrict scope deliberately to keep the work manageable and focused for practical reference use.

Who It's For

This work is aimed at researchers, graduate students, and instructors in pure mathematics who need a compact reference for group presentations and low-order group data. It is particularly handy for those working in algebraic or geometric group theory who require quick access to definitions and tables for permutation and reflection-generated groups.

It is less suitable as a first course textbook for beginners: readers seeking extensive pedagogical exposition, worked exercises, or a broad survey of all finitely generated groups should look elsewhere. The book assumes some familiarity with group-theoretic language and classical references.

Pros & Cons

Pros

• Concise tables provide immediate reference for low-order and classical groups.
• Useful treatment of reflection-generated groups for geometric applications.
• Compact focus makes it easy to consult specific group definitions without excessive background material.
• Includes material on symmetric, alternating, and linear fractional groups, aiding classification tasks.

Cons

• Not intended as an introductory textbook; readers new to group theory may find it terse.
• Scope is deliberately limited, so it does not catalog every conceivable finitely generated group.

Specifications

Title	Generators and Relations for Discrete Groups
Series	Ergebnisse der Mathematik und Ihrer Grenzgebiete. 1. Folge, 14
Authors	Harold Scott Macdonald Coxeter, William O. J. Moser
Main focus	Tables and definitions for finitely generated and low-order groups
Notable content	Permutation groups, reflection groups, linear fractional groups
Intended audience	Researchers and advanced students in pure mathematics

Our Verdict

Generators and Relations for Discrete Groups is a focused, practical reference that rewards readers who already know group-theoretic basics. It delivers compact, well-organized tables and concise treatments of reflection and permutation groups, making it good value for specialists who need reliable, quick access to classical group presentations.

Frequently Asked Questions

Does this book include tables for small permutation groups?
Yes, it contains tables and commentary on permutation groups of low degree and refers to classical exhaustive tables for degree up to 8.

Is this suitable for undergraduate self-study?
Not as a primary textbook; it is best used alongside more introductory texts because it is concise and assumes prior familiarity with group theory.

Are geometric reflection groups covered?
Yes, groups generated by reflections in real Euclidean space are discussed and tabulated for various dimensions.