The influence of different gomomorphic images on the structure of a group is one of the most important and natural problems of group theory. The problem of describing a group with all its gomomorphic images known, i.e. reconstructing the whole thing using its reflections, seems especially natural and promising. This theme has a history that is almost a half-century long. The authors of this book present well-established results as well as newer, contemporary achievements in this area from the common integral point of view. This view is based on the implementation of module theory for solving group problems. Evidently, this approach requires investigation of some specific types of modules: infinite simple modules and just infinite modules (note that every infinite noetherian module has either an infinite simple factor-module or a just infinite factor-module). This book will therefore be useful for group theorists as well as ring and module theorists. Also, the level, style, and presentation make the book easily accessible to graduate students.