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Preface Preface

This text is designed for a one-semester undergraduate mathematics course that combines an introduction to group theory with an introduction to modern geometries using the Kleinian paradigm. The topics overlap naturally: group theory begins with geometric symmetries like the dihedral groups, and geometry places transformation groups overtly in the forefront of geometric structure.

The chapter on groups develops the basic vocabulary and theory of groups and homomorphisms, culminating with group actions. The chapter on geometry makes use of group symmetries to build the basic theory of Möbius, hyperbolic, elliptic, and projective geometries. Throughout, a design theme is the use of a small but carefully chosen collection of tools, beginning with the algebra and geometry of complex numbers and quaternions, and using a minimum of machinery from analysis and linear algebra, to develop useful and nontrivial results for group theory and geometry in a way that prefers using conceptual tools to brute computation.

Many results (for example, Lagrange's Theorem for groups and area formulas in hyperbolic and elliptic geometries) are developed in carefully structured exercises, rather than in the reading. This reflects a deliberate emphasis on active engagement with the material. The intention is for students to read, to reason, and to develop results on their own, as a means of achieving mastery and fostering analytic skills.

The text assumes prerequisite courses in calculus, linear algebra, and experience with proof writing.

Here is an example of a possible schedule for a 15-week semester.

Ch. 1 Preliminaries 1.1 -- 1.4, Ch 1 exam 3 weeks
Ch. 2 Groups 2.1 -- 2.5, Ch 2 exam 6 weeks
Ch. 3 Geometries 3.1 -- 3.5, Ch 3 exam 6 weeks

Variations on this basic course are supported by "Additional Exercises" sections in Chapters 2 and 3, and a "Further Topics" section at the end of the text.