Explore the historical development of group theory in this comprehensive 59-minute lecture. Trace the evolution of this mathematical field from its roots in number theory and algebra to its expansion into geometry during the 19th century. Learn about Euler's work on Fermat's little theorem, Gauss' composition of quadratic forms, and the role of permutations in solving polynomial equations. Examine the symmetric group S_3 and discover how groups of transformations became linked to geometric symmetries through the work of Klein and Lie. Investigate the symmetry groups of Platonic solids as a bridge between algebraic and geometric aspects of group theory. Gain insights into key concepts such as Lagrange's theorem and polyhedral groups, making this lecture accessible even to those new to the subject.