Groups whose non-permutable subgroups satisfy certain conditions

Abstract

In this dissertation, we determine the structure of groups whose non-permutable subgroups satisfy certain conditions. In Chapter 1, we give the definitions and well-known results that we will use during the dissertation. In Chapter 2, we express our main result, which states that an infinite rank $mathfrak{X}$-group with all proper subgroups permutable or of finite rank has all subgroups permutable. Before proving our main result in Chapter 4, we establish some preliminary results in Chapter 3 which are used in proving the main result. In Final Chapter, we study the class of locally graded groups with all subgroups permutable or nilpotent of bounded class $c$. We prove that such groups are soluble of derived length bounded by a number depending on $c$. This chapter contains preliminary investigations into the problem of the structure of groups with all subgroups permutable or soluble. (Published By University of Alabama Libraries