The Chermak-Delgado Lattices Of Finite Groups

Suppose that G is a finite group,and H is a subgroup of G.The Chermak-Delgado measure of H is denoted by mG(H),and defined as mG(H)=|H|·|Cg(H)|.The subgroups of G that satisfy the maximal Chermak-Delgado measure form the sublattice of a subgroup lattice of G,called the Chermak-Delgado lattice of G.In recent years,the ChermakDelgado lattice has attracted the attention of well-known mathematicians Glauberman,Isaacs and Brewster et al.,and they made a contribution to Chermak-Delgado lattice.In this thesis,first we survey the research work on Chermak-Delgado lattice.Based on the knowledge,we determine the Chermak-Delgado lattice for finite p-groups with a large abelian subgroups of index p3,and determine the Chermak-Delgado lattice for 3-groups of maximal class and 5-groups of maximal class in which the least element of ChermakDelgado lattice is center,moreover,we describe the Chermak-Delgado lattice for general p-groups of maximal class.In addition,the Chermak-Delgado lattice of metacyclic pgroups is also determined.Our work can be regarded as an expansion of the work of Mcculloch and Tǎrnǎuceanu in[Comm.Algebra,48:1(2020),37-44].This thesis consists of four chapters.Chapter one is an introduction.Chapter two is a preliminary.In Chapter three,we survey the research work on Chermak-Delgado lattice.Chapter four consists of three sections.In Section one,the Chermak-Delgado lattice of metacyclic p-groups is determined.In Section two,we determine the Chermak-Delgado lattice for finite p-groups with a large abelian subgroups of index p3.In Section three,we determine the Chermak-Delgado lattice for 3-groups of maximal class and 5-groups of maximal class in which the least element of Chermak-Delgado lattice is center,moreover,we also describe the Chermak-Delgado lattice for general p-groups of maximal class.