Generalization of Sylow’s Theorem

Abstract

In the study of group theory, it is common to break up a complex group into simpler subgroups in order to arrive at a structure that is easier to analyze and understand. It is also sometimes possible to reconstruct the original group from these subgroups. Although this is not always possible, we can apply this process to finite solvable groups and derive some theorems regarding these groups. Sylow's theorem and Hall's theorem are among the most famous results. Hall's theorem, which is regarded as an extension of Sylow's theorem, states that if a group G is solvable and is of some order mn, where m is prime to n, then G has a subgroup of order m and all subgroups of this order are conjugate. When p = π, a Hall π-subgroup is simply a Sylow p-subgroup. While Sylow's theorem is valid for any finite group, Hall subgroups need not exist in nonsolvable groups. For example, A5 has order 60 = 3 · 20, but it has no subgroups of order 20. This is demonstrated within the paper. Hall's theorem has been the starting point for the theory of finite solvable groups developed over the past seventy years, although those results are not given here.