This book is devoted to examining a range of results and related research stemming from the following classical theorem: If the center of a group has finite index, then the derived subgroup of the group is finite.This theorem, long regarded as a cornerstone of group theory, has played a significant role in the development of infinite group theory. The body of work in which it appears is extensive. It has had a profound influence on the field, serving as a catalyst for numerous results and inspiring new approaches to the study of infinite groups. These developments have helped shape the branch of infinite group theory now commonly referred to as the "classical" theory.