The Counting Problems Of Group Theory And Hamiltonian Cayley Graphs

Arguing from the view of the group theory, we have achieved that the sub-group is closely related to the group structure. The subgroup has an effect to the group structure, and the group structure in its turn adds to the subgroup:the growth of knowledge follows a kind of compound interest law. Initially, this the-sis studies the counting problems of subgroups of p-group and considers the special condition of p= 2. It is well known that 2 is the only one even prime number, so that makes the structure of 2-group to be multifarious. Moreover, we put our hand to study the counting problem of subdomain of Galois field by some previous conclusions. It shows that the subdomain of Galois field has the similar properties as the subgroup of p- group. Eventually, according to the structure of groups of order pq~2 and the supersolvable group theory, we prove that every con-nected Cayley digraph of order pq~2 is Hamiltonian and that some of the Cayley graphs of order pq~2 are Hamiltonian.In the third chapter of this thesis, we use mainly the theorems of classification of 2-group and the count principles of P.Hall to put an end to the problem about the counting of subgroups of 2-group. Then we establish the relationship between Galois field and Galois group by the classical theory of Galois. Finally we find some beneficial properties of the subdomain of Galois field.In the fourth chapter of of this thesis, we firstly look back the progresses of the hamiltonian Cayley graphs. With the help of the theorems of the group structure and the supersolvable group, we solve the challenge that every connected Cayley graph of order pq~2 is Hamiltonian. Then, we take a deeply study on the connected Cayley graphs of order 3p. Finally, we hope that two examples can open the subject for discussion.