Generalized Dedekind Groups

In the finite group theory,it is quite interesting to investigate the structure of finite groups if normal subgroups are of some kind of properties or normalizers and centralizers of subgroups satisfy a certain relationship.Lots of important and significant results have been given.In this thesis our motivation come from the following two basic facts.If A is a subgroup of a finite group G with A ≥ or A ≤ Z(G),then A(?)3G;If A is an abelian subgroup of a finite group G,then A≤ CO(A)≤ NG(A)≤G.A finite p-group G is said to be a CCts-group if |G’/(G’∩N)]≤ ps or|N/(N∩ Z(G))| ≤ pt for every normal subgroup N of G,where s and t are non-negative integers.Obviously,if G is a CC00-group,then G’ ≤ N or N ≤ Z(G).In chapter III,we investigate the structure of CCts-groups.We first prove that the order of G’Z(G)/Z(G)is bounded by ps+t+1 and the upper bounds of the exponent of G’ with G’≤ Z(G)is ps+t+1 for a CCts-group G.Then we try to give some elementary properties of p-groups with very small derived subgroups by using the properties of capable p-groups.We also describe the structure of CC11-groups.A finite p-group G is called a CC-group if G’/G’ ∩ N is cyclic or N/N ∩ Z(G)is cyclic for every normal subgroup N in G.Obviously,CC-groups is the pro-motion of CC11-groups.In chapter IV,we discuss the structure of CC-groups.We first investigate the properties of CC-groups,and then we give a necessary and sufficient condition for a p group G to be a CC-group if the quotient group G/Z(G)is generated by two elements.A finite group G is called a NNC-group if for any non-normal abelian sub-group A,either NG(A)= CG(A)or CG(A)= A.In chapter V,we study the structure of NNC-groups.We provide a complete classification of nilpotent NNC-groups and non-solvable NNC-groups.We also investigate the solvable NNC-groups,and manage to describe the structure of solvable NNC-groups.