The Number And Group Structure Of Conjugate Classes And The Same Order Subgroups

The quantitative characterization is the investigation of the influence of its quantitative information on the structure and property of finite group. This disser-tation continues to investigate the topic with quantitative information of the order, the conjugacy class size, the number of subgroups of possible order of finite group.In Chapter1, we introduce commonly used symbols, terminology and con-cepts, and also the research background and main results in this thesis.In Chapter2, we investigate the effect of the set of its conjugacy class sizes on the structure of almost simple group, which is a directly extensive subject of Thompson conjecture.In1987, John G. Thompson conjectured that if G is a finite group with Z(G)=1and L is a simple group satisfying that N(G)=N(L), then G≌L, where N(G) is the set of conjugacy class sizes of G. In fact, this conjecture is a quantitative characterization for non-abelian simple group with its conjugacy class sizes. In view of the similar between simple group and almost simple group, we hope that we can characterize some almost simple with their conjugacy class sizes. In this chapter, we prove that the almost sporadic simple groups Aut(McL) and Aut(J2). linear groups PGL3(4) and PGL3(7), can be determined by the set of their conjugacy class sizes. Moreover, the prime graph of these groups are connected. Therefore, the validity of Thompson conjecture is generalized to these almost simple groups.In Chapter3. we investigate the influence of the order and the special con-jugacy class sizes on the structure of (almost) simple groups, which is a indirectly extensive subject of Thompson conjecture. Recently, Thompson conjecture has en-couraging progress. But still it is not completed. It has been proved that the conjecture holds for all the simple groups with non-connected prime graph and par-tial simple groups with connected prime graph. Therefore, during the investigation of this subject, we only use some special conjugacy class sizes(as few as possible) and assumed|G|=|L|to successfully characterize the automorphism groups of sporadic simple groups, PSL2(p), PGL2(p), and simple K4-groups. Note that we do not use the classification theorem of finite simple groups in the proof of PSL2(p) and PGL2(p). As a consequence of these results, the validity of Thompson conjecture is generalized to these (almost) simple groups. In Chapter4.we study the structure and property of fmite group G with n(G)={1,m}.We obtain the following results:The nilpotent length of G is less than2.If G is a nilpotent group,then m=p+1or p2+p+1,p∈π(G).and then we classify such kinds of groups.Further.we also classfy the non-nilpotent groups G with m=p+1.