Certain Limited Almost One Group The Number Of Properties And Group Structure Research

The topic of this dissertation is the investigation of the influence of the degree pattern related to its prime graph and the large irreducible character degrees on the structure of finite groups.In Chapter1, we introduce the research background and main results in this thesis.In Chapter2, we study the effect of the degree pattern related to its prime graph on the structure of finite groups, especially almost simple groups.Let G be a finite group and|G|=pα11pα22…pαkk,where pi are primes and αi are integers. For p∈π(G), let deg(p):=|{q∈π(G)|p～q}|, called the degree of p. We also define D(G):=(deg(pi),deg(p2),…, deg(pk)), where p1<p2<…<Pk. We call D(G) the degree pattern related to prime graph of G. In2005, A. R. Moghaddamfar, A. R. Zokayi, M. R. Darafsheh. etc. first put forward that if G is a finite group, then G can be characterized by its order|G|and the degree pattern D(G), and for the first time successfully characterize the sporadic simple groups, the alternating groups Ap、Ap+1、Ap+2, the symmetric groups Sp and Sp+1, some Lie type simple groups L2(q), L3(q), U3(q),2B2(q),2G2(q), all finite simple C2,2-groups and all the automorphism groups of sporadic simple groups, etc., where p is a prime and q is a prime power (see [3,4,7,8]). In this thesis, we continue this investigation, and show that the almost simple K3-groups, almost orthogonal groups O10±(2), the alternating groups Ap+3、Ap+5、Ap+7, the symmetry groups Sp+3、Sp+5and Sp+7, etc., where p is certain prime and p>100, can also be characterized by their orders and the degree patterns. In the end of this chapter, we put forward two open problem related to the topic.In Chapter3, we study the influence of the large irreducible character degrees on the structure of finite groups, especially almost simple groups.In2000, Huppert conjectured that if H is nonabelian simple group, and G a group such that cd(G)=cd(H), then G=H×A, where A is an abelian group. Huppert point out that each finite non-abelian simple group G is characterized by cd(G), the set of degrees of its complex irreducible characters, and he confirmed that the conjecture holds for simple groups such as L2(q) and Sz(q). Moreover, he also proved this conjecture holds for19out of26sporadic simple groups, and a few others simple groups (see [48]-[51]). We mention that Huppert conjecture is still unsolved completely. By the way, we see that we must consider all the irreducible characters of G in this conjecture, so the conditions are very strong. Here we want to weaken this condition and consider only some special irreducible characters, such as the largest irreducible characters, the second largest irreducible characters or the third largest irreducible characters and so on. In this dissertation, we characterize the automorphism groups of simple.K3-groups, some sporadic simple groups, some simple K-groups, the automorphism groups of Mathieu groups, and other simple groups such as L5(2), O8+(3), R(27),L2(29) and S8(2) and so on. In a sense, We weaken essentially the condition of Huppert’s conjecture for the groups considered as above in this chapter.