The Influence Of The Subgroups With Given Order On The Structure Of Finite Groups

In this paper, wo main study the influence of the subgroups with given order on the structure of finite groups, and then we show the detailed characterization of solubility、p-nilpotency and p-supersolubility of finite groups, furthermore, we revealed the structure of generalized hypercentre of finite groups. We divide the dissertation into five chapters.Chapter 1, The introduction. In this chapter we introduce the research back-ground and the main results of the dissertation.Chapter 2. The basic definition. In this chapter we give some basic definitions of group theory contained in this dissertation.Chapter 3, The influence of Mp-supplemented subgroups with certain order on the structure of finite groups. In this chapter, firstly, recall the main results of cite [55] of Monakhov. Then we show the more general results of cite [55]. That is, for different primes, we obtain the p-nilpotency and p-supersolubility of group G by using the Mp-supplementation of certain subgroups of H with order π. For example, let G be a p-solvable group and p be a prime divisor of |G|. Suppose D is a subgroup of FP(G) containing Op’(G) and Dp≠1. If every subgroup T of Fp(G) with |T|=|D| is Mp-supplemented in G, then G is p-supersolvable. For the prime 5, we have the structure of composition factor L/K of G. Let H be a 5-nilpotent subgroup containing a Sylow 5-subgroup P of G. Suppose H has a subgroup D with D5≠1 and |H:D|=5α. If every subgroup T of H with |T|=|D| is M5-supplemented in G, then every composition factor E/K of G satisfies one of the following conditions:(1) E/K is cyclic with order 5;(2) E/K is a 5’-subgroup;(3) E/K is isomorphic to A5.This is also one of the highlights of this chapter. Furthermore, according to the Frobenius’s Theorem, we obtain the sufficient and necessary conditions for the p- nilpotency of G. In particular, we obtain the solubility of G by using the Mp-supplementation of certain subgroups. Let π={3,5}. For any p∈π, suppose that P is a Sylow p-subgroup of G and H is a p-nilpotent subgroup of G which containing P. If H has a subgroup D with Dp≠1,|H:D|=pa, and every subgroup T of H with|T|=|D| is Mp-supplemented in G, then G is solvableChapter 4, The influence of the local properties of certain subgroups on the generalized hypercentre. In this chapter, we study the structure of the generalized hypercentre of finite group G by using.M-supplemented and.Mp-supplemented subgroups with given order. For example, if E is a normal subgroup of G and some certain subgroups of E is M-supplemented in G, then every G-chief factor below E is cyclic. Then the relevant results are applied to the solubly saturated formations. For example, let E be a normal subgroup of G such that G/E is p-quasisupersoluble. Suppose that every noncyclic Sylow subgroup P of X has a subgroup D such that 1<|D|<|P| and every subgroup H of P with order|H|=|D|is M-supplemented in G, where X=E or X=F*(E). Then G is p-quasisupersoluble.Chapter 5, The influence of m-embedded subgroups on the structure of finite groups. In this chapter, we study the structure of finite groups by using m-embedded subgroups and obtain some new results about p-supersolubility and p-nilpotency of finite groups. Particularly, we suggest two new criterions for the p-nilpotency of G. That is, let p be an odd prime divisor of |G| and P be a Sylow p-subgroup of G. Suppose that every maximal subgroup P1 of P is m-embedded in G. Then G is p-nilpotent if one of the following conditions holds:(1) NG(P1) is p-nilpotent for every maximal subgroup P1 of P.(2) NG(P) is p-nilpotent.Furthermore, we have the following result:Let G be a p-solvable group and P a Sylow p-subgroup of G. If every maximal subgroup of P is m-embedded in G, then G is p-supersolvable. This chapter is a supplementation of Theorem 4.1 of cite [16].