Two directions in the development of the theory of non-simple finite groups

In this article we present some biographical material on P. L. M. Sylow, prove his theorem by Wielandt's method, and give a survey of the most important generalizations of the theorem that have appeared since its discovery 100 years ago. We note that the main result in Gaschьtz' first paper on the theory of formations is a generalization of Sylow's theorem in the class of finite soluble groups. We list the objectives of this theory and state the results obtained over the first decade of its development.