Sylow and Hall Theory

This paper presents Phillip Hall's theory on subgroups as a generalization of the Sylow theory on subgroups. As introductory material, we initially develop the fundamental aspects of group theory, including discussions of groups, subgroups, and particular theorems and lemmas which define the relationship between groups and their subgroups (LaGrange, Cauchy). Following this cursory sketch of group theory, we introduce and prove the Sylow theorems, giving examples of various Sylow p-subgroups of some elementary groups to help elucidate Sylow theory. We further present abstract examples\ud of groups of order p, p2, pq, and p3 , following which we define Hall subgroups and compare Hall and Sylow theory on subgroups