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.

David W. Lyons is a professor of mathematics at Lebanon Valley College in Annville, Pennsylvania, USA, where he has led a student-faculty research program in quantum information science since 2002.

---