Explore the fascinating history of group theory and its evolution from number theory to describing geometric symmetry in this comprehensive mathematics lecture. Delve into the search for simple groups, symmetry in various types of tessellations, and the introduction of Lie groups by Sophus Lie. Discover remarkable mathematical objects like the Golay code, Mathieu groups, and exceptional Lie groups, culminating in the Monster group. Learn about the classification of finite simple groups, a pinnacle achievement in 20th-century mathematics, while gaining insights into polygons, frieze groups, finite simple groups, and projective linear groups.