Explore the concept of isometry groups in projective geometry through this 34-minute lecture from the Rational Geometry Math Foundations series. Delve into the Euclidean structure of the projective line and examine its symmetries, focusing on rotations and reflections as the primary types of isometries. Discover how these symmetries form a group structure, providing a concrete introduction to group theory. Learn to derive algebraic relationships between isometries and gain insights into the mathematical foundations of symmetry. Ideal for those interested in advanced geometry, group theory, and the intersection of abstract mathematics with concrete applications.