Several Problems Of Graph And Group

In this thesis,we study some problems on groups and graphs.The main results include three parts:Cayley graphs of small valency,cubic semisymmetric graphs and the influence of its generalized normal subgroups on the structure of the group.A Cayley graph X=Cay(G,S) of group G is said to be normal if R(G),the group of right multiplications,is normal in AutX.In Chapter Two of this paper, by investigating the normality,we classify cubic Cayley graphs of dihedral group of order 4p~2,where p is an odd prime.In addition,we obtain two GRR and a family of non-normal Cayley graph.Also,we solve several isomorphism problems of Cayley graph of semidihedral groups whose normality were determined.If a subgroup G of AutX acts transitively on V(X)(or E(X)),we say that X is G-vertex-transitive(or G-edge-transitive).In the special case when G= AutX, we say that X is vertex-transitive(or edge-transitive).If AutX acts transitively on A(X),we say that X is arc-transitive or symmetric.A regular edge-transitive but not vertex-transitive graph will be referred to as a semisymmetric graph.In Chapter Three of this paper,we classify all connected semisymmetric cubic graphs of order 6pq,where p and q are odd primes with 3＜p＜q.A subgroup of G is S-quasinormal in G,if it is permutes with every Sylow subgroup of G.In[74],the authors gave a characterization of a finite group G under the assumption that every subgroup of the generalized Fitting subgroup of prime order is S-quasinormal in G.In Chapter Four of this paper,firstly,we provide two counterexamples of this result,and make a correction to it;secondly,we extend the result under the assumption that every subgroup of generalized Fitting subgroup of N of prime order is S-quasinormal in G,where N is a normal subgroup of G such that G/N is supersolvable;as a byproduct,we get some properties of weakly S-supplemented subgroup and C_Δ-supplemented subgroup.