The wreath product.

Abstract

This thesis deals with a topic in abstract algebra, the wreath product. The wreath product is a special type of permutation group which acts on ordered pairs. An example is given to illustrate the algebraic structure of the wreath product. methods of performing the operation of composition of mappings as defined for the wreath product are demonstrated. Theorems concerned with the structure of wreath products are developed. The importance of the concept 9f wreath products lies in their use in constructing certain this of subgroups of symmetric groups. These subgroups are the Sylow p-subgroups of symmetric groups. The method of constructing Sylow p-subgroups with wreath products is developed. Computation of the number of Sylow 3-subgroups of the symmetric group on thirteen elements is performed. Similar computations for symmetric groups on 12, 14, and 15 elements are shown. One chapter is devoted to investigating which wreath products have the same internal structure; that is, which are isomorphic. Theorems demonstrating isomorphisms between certain wreath products with the same number of elemcnts, that is, the SaJ:JC order) are developed, and conclusions for wreath products of order less than 100 are derived from these theorems. Some minor results of the study are presented in Chapter VI.