Subnormality Of Normal Edge-transitive Cayley Graphs

Groups and graphs have always been the central objects of mathematical research.Combining them to study graphs by using the theory of groups,or to study or represent groups by using graphs,have promoted the development of algebraic graph theory.In1938,R.Frucht proved that for any given abstract group,there exists a graph which admits the group to be its automorphism group.This important work opened the curtain of this field.However,extensive research in this field was carried out after 1960.In the past 30 years,there have been many important work in this field,such as studying the application of graph theory in group theory,especially using graph theory to study permutation groups,for example to study the structure of the suborbit of certain primitive groups.Some simple groups were found as automorphism groups of graphs,making an important contribution to the classification of finite simple groups.On the other hand,the research of applying group theory to graph theory has had richer results in the last 30 years.One of the main research directions in algebraic graph theory is the study of Symmetric Graphs,and one of the important problems in the study of symmetric graphs is the study of various Cayley graphs.An important problem in the study of Cayley graphs is the study of normal Cayley graphs,which has been extended to subnormal cases in recent two years.The main purpose of this paper is to study the connected subnormal edge-transitive Cayley graphs on some groups.Cayley graphs on different groups have different properties.The groups mainly considered in this paper are Frobenius groups whose order is the product of two prime numbers and non-abelian groups of order 4.These groups and their automorphism groups have specific properties.Therefore,in this paper,we first introduce some basic properties of these groups,and then obtain their automorphism groups according to their properties.On this basis,we further study the subnormal edge-transitive Cayley graphs on these groups.Finally,it is proved that the normal edge-transitive Cayley graphs on these groups are either normal or subnormal.Another result of this paper is to study subnormal connected 2-path transitive Cayley graphs.For this problem,we characterize all subnormal 2-path transitive Cayley,and prove that it is either normal or the normal cover of complete bipartite graphs.