The concept of a group is as old as mathematics itself. However, it entered mathematical consciousness only at the beginning of the 19th century. The requirements of Galois' theory of algebraic equations led to the development of a theory of finite permutation groups, which evolved over the course of a century into a broad theory of groups of finite order. During the same period, the trend of invariant theory in geometry and analysis also prompted the investigation of special infinite groups, leading to further expansion of group theory. Conversely, the development of this theory provided new methods in almost all areas of mathematics. This interplay of mutual inspirations gave rise around 1920, coinciding with the complete structural transformation of algebra and the advent of set theory.