Introduction to Groups and Geometries

Status update 11/4/2020

The first edition draft is complete! Editing continues, with a stable edition target date of early January 2021. Discussion is welcome! Contact the author at lyons (at)

About the Textbook

This free, online 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 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.