The Wreath Product of Powerful p-Groups

This study provides a scholarly examination of fundamental concepts within the field of group theory, specifically focusing on topics such as the wreath product and powerful p-groups. We examine the characteristics pertaining to the structure of the wreath product of cyclic p-groups, with a specific focus on the groups that are powerfully embedded within it. The primary discovery pertains to the construction of the powerful wreath product and the quasi-powerful wreath product. In this study, we establish that subgroups are powerful within the wreath product, specifically focusing on p-groups. The aforementioned outcome is derived from the assumption that p is a prime number and W is the standard wreath product of two nontrivial cyclic p-groups, denoted as G and H.