The Sylow theorems and their generalizations

Thesis (M.A.)--Boston UniversityThe thesis treats the generalization of Sylow's theorems about p-subgroups to theorems concerning Hall π-subgroups. The Hall π-subgroup is an extension of the Sylow p-subgroup to a set of primes π instead of the single prime p. ln the first section Sylow's theorems are stated and the Hall π-subgroup is defined. Hall [3] generalized Sylow's theorems completely in the case of solvable groups. ln particular he showed that every solvable group possesses a Hall π-subgroup for any set of primes. He also showed [4] that if a group G is not solvable there is at least one set of primes π such that G possesses no Hall π-subgroup. His results are discussed in section 2 and some examples of insolvable groups for which his generalized theorems are invalid are given. For a particular set of primes π it is possible to generalize Sylow's theorems even when the groups being considered are not necessarily solvable. Some theorems of this type are considered in section 3. For example, one of the theorems gives sufficient conditions, depending on the set of primes π, a group to possess a Hall π-subgroup. For a particular set of primes π, solvability can be generalized to π-separability and π-solvability. Theorems similar to Hall's theorems for solvable groups can be proved for π-separable and π-solvable groups. These are discussed in section 4. A theorem concerning sufficient conditions for a group to be π-separable is proved. Finally in section 5, some of the results of the earlier sections are applied to the problem of subnormal subgroups and their normalizers