# Introduction to Groups and Geometries

## Status update 5/12/2023

*The May 2023 edition has been classroom tested for three years
running in a semester-long
course at Lebanon Valley College. Discussion is
welcome! Contact the author at lyons (at) lvc.edu.*

## 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 MÃ¶bius, 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.

quantum.lvc.edu/lyons