Subgroup Lattices Of Finite Groups And Their Combinatorial Counting Problems

The subgroup lattices of groups is not only a perfect bridge between group-s and lattices,the two basic algebraic systems,but also an important research object in group theory.Many famous group theorists have been engaged in this research field and achieved fruitful results.In this thesis,we mainly study the relation of the partial order structures of subgroup lattices and the group struc-tures,and the combinatorial counting problems of subgroup chains of subgroup lattices for finite groups.The former part adopts the topology methods,while the latter part has important applications in the theory of fuzzy groups.The backgrounds and progresses on these two research fields are presented in intro-duction,the chapter ?.In chapter ?,we recall some definitions and conclusions on subgroup lattices and topologies in order to apply them for the convenience of follow-up research.In chapter ?,we study the relations between homotopy equivalence prop-erties of subgroup lattices and the group structures for finite groups.Consid-er two subposets of the subgroup lattices L(G)of a finite group G:Sp(G)={all nontrivial p-subgroups of G} and Ap(G)={all nontrivial elementary abelian p-subgroups of G}.Viewed as finite topological spaces,assuming that Sp(G)is contractible,we obtain the sufficient conditions such that A-(G)is contractible as well.On the other hand,we study the homotopy types of L(G)-{1,G} and obtain that if G is a nontrivial finite nilpotent group other than prime order,then L(G)-{1,G} is contractible if and only if ?(G)?1.Furthermore,we describe the homotopy types of L(G)—{1,G} when the finite group G is simple,dihedral and generalized quaternion.In chapter ?,we study the combinatorial counting problems of subgroup chains of subgroup lattices of finite groups.Considering a finite group G=A×B with(|A|,|B|)=1,we build a one to one correspondence between a set of subgroup chains of G and a set of vectors on positive integers.Therefore the counting problem of the subgroup chains of such a finite group is transformed into the counting problem of vectors with specific properties.In this way,we obtain the formula on calculating the number of subgroup chains of G and reduce the counting problem of G to the problems of A and B.As some applications of this result,we give the formulas of subgroup chains of some classes of finite groups such as abelian groups and nilpotent groups.These results greatly propel the previous research and make a big step on the combinatorial counting problems of subgroup chains.