Guo Xiuyun

C -Supplemented subgroups of finite groups

A subgroup H of a group G is said to be c-supplemented in G if there exists a subgroup K of G such that HKa G and H\ K is contained in CoreGOHU .W e follow Halls ideas to characterize the structure of the finite groups in which every subgroup is c-supplemented. Properties of c-supplemented subgroups are also applied to determine the structure of some finite groups.

The influence of c-normality of subgroups on the structure of finite groups

Abstract A subgroup H of a finite group G is said to be c-normal in G if there is a normal subgroup N of G such that HN=G and H∩N≤HG=coreG(H). In this paper we investigate further the influence of c-normality of some subgroups on the structure of finite groups and generalize some known results.

Cover-avoidance properties and the structure of finite groups

Abstract We call a subgroup A of a finite group G a CAP -subgroup of G if for any chief factor H/K of G , we have H∩A=K∩A or HA=KA . In this paper, some characterizations for a finite group to be solvable are obtained under the assumption that some of its maximal subgroups or 2-maximal subgroups be CAP -subgroups. We also determine the p -solvability and p -nilpotency of finite groups by considering their CAP -subgroups.

The influence of c-normality of subgroups on the structure of finite groups II *

A subgroup Hof a finite group Gis said to be c-normal in Gif there exists a normal subgroup Nof Gsuch that HN=Gand H∩ N≤ H G =core G(H).In this paper we shall investigate the influence of c-normality of some subgroups on the strcture of finite groups further, and obtain some results on some kinds of weaker conditions.

On theta pairs for a maximal subgroup

For a maximal eubgroup M of a finite group G, a 8-pair is any pair of subgroups (C,D) of G such that (i) D▹G, D≤C, (ii) - G, - M and (iii) C/D has no proper normal subgroup of G/D. A partial order may be defined on the family of 8-pairs. Let △(M) - {(C,D)|(C,D) is a maximal 8-pair and CM - G}. The purpose of this note is to prove: (1) A group G is solvable if and only if, for each maximal subgroup M of G, △(M) contains a 8-pair (C,D) such that C/D ie nilpctent. (2) If a group G is S4-free, then G ia eupersolvable if and only if, for each maximal subgroup M of G, △(M) contains a 8-pair (C,D) auch that C/D is cyclic

On complemented subgroups of finite groups

A subgroup H of a finite group G is said to be complemented in G if there exists a subgroup K of G that G=HK and H∩K=1. In this case, K is called a complement of H in G. In this note some results on complemented subgroups of finite groups are obtained.

ON THE DESKINS INDEX COMPLEX OF A MAXIMAL SUBGROUP OF A FINITE GROUP

Let M be a maximal subgroup of a finite group G. A subgroup C of G is said to be a completion of M in G if C is not contained in M while every proper subgroup of C which is normal in G is contained in M. The set, I(M), of all completions of M is called the index complex of M in G. Set P(M) = {C ϵ I(M) ¦ C} is maximal in I(M) and G = CM. The purpose of this note is to prove: A finite group G is solvable if and only if, for each maximal subgroup M of G, P(M) contains element C with CK(C) nilpotent.

Permutability of minimal subgroups andp-nilpotency of finite groups

In this paper it is proved that ifp is a prime dividing the order of a groupG with (|G|,p − 1) = 1 andP a Sylowp-subgroup ofG, thenG isp-nilpotent if every subgroup ofP ∩GN of orderp is permutable inNG(P) and whenp = 2 either every cyclic subgroup ofP ∩GN of order 4 is permutable inNG(P) orP is quaternion-free. Some applications of this result are given.

On Deskins's conjecture

Abstract Let M be a maximal subgroup of a finite group G ; then a subgroup C of G is said to be a completion of M in G if C is not contained in M while every proper subgroup of C which is normal in G is contained in M. The set, I ( M ), of all completions of M is called the index complex of M in G . Set P ( M ) = C e I ( M ) ⊢ C is maximal in I ( M ) and G = CM . The purpose of this note is to prove: If a finite group G is S 4 -free, then G is supersolvable if and only if, for each maximal subgroup M of G, P ( M ) contains an element C with C/K ( C ) cyclic.

Normal π—complements in finite groups

The main result of this note is the following: theorem Let G be a finite group and let π be a set a promes. Then G has a normal π-complement if and only if the following three conditions are satisfied: (A) There exists a π-Hall subgroup H of G. (B1) If x∊H and then the number of π′-elements of CG(x) is . (C)