Influences Of The Centralier And S-Normality Of A Minimal Subgroup On The Structure Of A Finite Group

The thesis focuses on the centralizer and S-normality of a minimal subgroup of a finite group and aims at studying their influences on the structure of a finite group, such as solvability, p — solvability, nilpotency, p — nilpotency, and super-solvability of a finite group, etc. On the basis of these discussions, the paper first advances a new concept, an s~*— supplement subgroup of a finite group, and explores its inheritance of a subgroup and a quotient group. Moreover, the paper discusses the sufficient conditions of p— nilpotency and super-solvability of a finite group and extends them to a formation and obtains some new significant results, which is a pioneer research on the structure of a finite group.The body of the thesis, in accordance with the research focuses, is divided into four chapters:The first is concerned with the theretical and practical backgroud information for the current research.The second is to investigate the relationship between the eentralizer of a minimal subgroup and the structure of the corresponding finite group. In the literature [1] and [2], the eentralizer of a minimal subgroup has been applied to study the solvability of a finite group and p — solvability by P.Cuccia-M.Liotta and Li Shirong . We follow this tradition and continue studying the structure of a finite group from the point of view of the eentralizer of a minimal subgroup. The conclusions arc made as the followings, which develop the findings by P.Cuccia-M.Liotta and Li Shirong and enrich the study on the structure of a finite group.1. Let S(G) be the set of minimal subgroups of odd order of G which are complemented in G. For every minimal subgroup X of odd order of G which does not belong to S(G),assume that C_G(X) is either subnormal or abnormal in G. Then G solvable.2. Let p be a minimal odd prime divisor of G.For each subgroup of G of order p,if cither X is complemented in G or Cg(X) (?) G, Then G is p-solvable.3. Let p be a odd prime divisor of G.G solvablc,for each subgroup of G of order p,if one of the following condition is satisfied:(1) either X is s-quasinamal in G or C_G(X) (?)G.(2) either X is e-normal in G or C_g(X) (?)G.