The Influence Of Weakly HC-Subgroups On The Structure Of Finite Groups

Let G be a finite group. A subgroup H of G is called weakly normal in G if H9<NG(H) implies g∈NG(H) for all g in G; a subgroup H of. G is said to be weakly HC in G if there exists a normal subgroup N of G such that G=HN and H∩N is weakly normal in G. and H also is called a weakly HC-subgroup of G.Since the properties of subgroups are closely related to the structure of the group, so in this paper, by using the weakly HC property of some special subgroups (such as Sylow-subgroups, prime subgroups, cyclic subgroup of order4) of G, we obtain some sufficient and necessary conditions for a finite group to be p-nilpotent and supersolvable. Some related results are generalized.This paper is divided into two parts according to its contents.In the first chapter, we mainly introduce the investigative background of this paper, in-vestigate the relationships among C-normal subgroup,weakly normal subgroup,H-subgroup and weakly HC-subgroup, and give some definitions and lemmas. By these lemmas, we can obtain some properties and theorems about weakly HG-subgroups.In the second chapter, we mainly use the weakly HC property of cyclic subgroups of order4and prime subgroups of G, we obtain some sufficient and necessary conditions for a finite group to be p-nilpotent and supersolvable.some main results as follows:Lemma1.2.2Let N, K and H be subgroups of G.(1) If H≤K≤G, and H is a weakly HC-subgroup of G, then H is a weakly HC-subgroup of K as well.(2) If N≤G and N≤H, then H is a weakly HC-subgroup of G if and only if H/N is a weakly HC-subgroup of G/N. (3) If H is a p-subgroup of G which is a weakly HC-subgroup of G. and N is a normal p’-subgroup of G, then HN and HN/N are weakly HC-subgroups of G and G/N respectively.Theorem2.1.1Let p be the smallest prime dividing the order of G and let P be a Sylow р-subgroup of G. If all elements of р(P) are weakly HC-subgroups of G, then G is p-nilpotent.Theorem2.1.8Let G be a finite group. If p is a fixed prime number and all elements of рр(G) are contained in Z∞(G). In addition, if p=2, all elements of р4G) are weakly HC-subgroups of G or lie in Z∞(G), then G is p-nilpotent.Theorem2.1.13Suppose that N is a normal subgroup of G such that G/N is p-nilpotent, where p is a fixed prime number, and all elements of рр(N) are contained in Z∞(G) In addition, if p=2, all elements of р4{N) are weakly HC-subgroups of G or lie in), then G is p-nilpotent.Theorem2.1.18Let p be the smallest prime dividing the order of G and P is a Sylow р-subgroup of G. If all elements of рр(P∩G’) and all elements of р4(P∩G’) are weakly HC-subgroups of G, then G is p-nilpotent.Theorem2.1.19Let p€π(G) such that (|G|,p-1)=1, then G is p-nilpotent if and only if there exists a normal subgroup N of G such that G/N is p-nilpotent, all elements of рр(N) and all elements of р4(N) are weakly HC-subgroups of G.Theorem2.2.1Let G be a finite group. If all elements of р*(G) are weakly HC-subgroups of G, then G is supersolvable.Theorem2.2.2Suppose that N is a normal subgroup of G such that G/N is super-solvable. If all elements of р*(N) are weakly HC-subgroups of G, then G is supersolvable.Theorem2.2.3Suppose that N is a normal subgroup of G such that G/N is super-solvable. If for a Sylow р-subgroup of N, all elements of р*(P) are weakly HC-subgroups of G, then G is supersolvable.