The Structures Of Some Finite Groups Under The Restrictions Of Prime Graphs Of Conjugacy Class Length

The conjugacy class lengths of finite groups are closely related to the finite group structures.For the study of finite group structures and properties,this paper mainly starts from the point of view of prime graphs of conjugacy class lengths.The group structures are represented by prime graphs of conjugacy class lengths as the restrict conditions.Scholars have given some important conclusions,for example,when the diameter of prime graphs of conjugacy class length is equal to 3,G=H:N,H and N are two abelian groups satisfying（|H|,|N|）=1;if there is no complete vertex in the prime graph of conjugacy class lengths,then G-KH,K<G,the order of K and H are primes and they are all abelian groups,and K=G‘,K∩Z（G）=1;the conjugacy class lengths prime graphs have at most one complete vertex,G is solvable group,and G’≤F₂（G）;If conjugacy class lengths prime graphs are non-connected,its two connected branches are complete graphs,and such graphs are quasi-Frobenius groups with abelian kernels and complements.First,introducing the related concepts of finite groups,research status at home and abroad,the research background of conjugacy class lengths prime graphs,and discussing the influence of prime graphs on the finite group structures.Second,the finite group structures corresponding to some prime graphs are studied,the finite group structures of the special prime graphs are analyzed.Furthermore,motivated by related conclusions of regular graphs,the finite group structures corresponding to 2-Regular prime graphs of conjugacy class lengths are discussed,if G is a 2-regular graph with 4 vertices,then G is solvable group and G=（N:H）xK,which N is regular Hall-subgroup of G,H is abelian Hall-subgroup of G,| π（N）|=| π（H）|-2,K is central Hall-subgroup of the G,and（|N|,|H|）=（|N|,|K|）=（|K|,|H）=1.Moreover,the finite group structures corresponding to 2-Regular graphs of two connected branches are generalized.Combined with GAP,the complete graphs of 4 vertices within 2000 order and two connected branches graphs of 5 vertices within 10000 order are classified.At the same time,the finite group structures are obtained by restricting the number of complete vertices in the prime graphs,the number of connected branchs and the diameter of the prime graphs.Finally,this paper summarizes the existing results and make an outlook in the end.