Introduction to Groups and Geometries

Status update 5/14/2021

The January 2021 edition has been classroom tested in a semester-long course at Lebanon Valley College. The book is now ready for general use. Discussion is welcome! Contact the author at lyons (at)

About the Textbook

The purpose of this text is to take advantage of the overlap between introductory courses in group theory and modern geometry. Group theory typically features geometric content in the form of symmetry groups, and Kleinian geometry relies on the group structure of congruence transformations. Learning the two subjects together enhances both.

This text is two textbooks in one: an introduction to group theory, and an introduction to modern geometries using the Kleinian paradigm. The book can be used for a combined one-semester course in both subjects, or, through supplementary projects, it can be used for a one-semester introduction to group theory or a one-semester introduction to modern geometries.

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.

The text is designed for active engagement, with carefully structured exercises throughout.

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

About the Author

David W. Lyons is a professor of mathematics at Lebanon Valley College in Annville, Pennsylvania, USA, where he has taught and conducted research since 2000. Lyons works in mathematical physics, leading a student-faculty research program in quantum information science since 2002. For more information, visit his academic website at the URL below.