Let G be a finite group. For a finite p-group P the subgroup generated by all elements of order p is denoted by Ω1(p). Zhang  proved that if P is a Sylow p-subgroup of G, Ω1(P) ≦ Z(P) and NG(Z(P)) has a normal p-complement, then G has a normal p-complement. The object of this paper is to generalize this result.
Two sufficient conditions for a finite group G to be p-supersolvable have been obtained. For example (Theorem 1.1), let N be a normal subgroup of G such that G/N is p-supersolvable for a fixed odd prime p and let Np be a Sylow p-subgroup of N. Suppose that N is p-solvable and Ω1(Np) is generated by the subgroups of order p of Np which are normal in NG(Np). Then G is p-supersolvable.
Let G be a finite group. A PT-group is a group G whose subnormal subgroups are all permutable in G. A PST-group is a group G whose subnormal subgroups are all S-permutable in G. We say that G is a PTo-group (respectively, a PSTo-group) if its Frattini quotient group G/Φ(G) is a PT-group (respectively, a PST-group). In this paper, we determine the structure of minimal non-PTo-groups and minimal non-PSTo-groups.
This paper represents an attempt to extend and improve the following result of Berkovich: Let G be a group of odd order. Let G=G1G2 such that G1 and G2 are subgroups of G. If the Sylow p-subgroups of G1 and of G2 are cyclic, then G is p-supersolvable.
Authors:Khaled A. Al-Sharo and Ibrahim A. I. Suleiman
A subgroup H of G is c-permutable in G if there exists a permutable subgroup P of G such that HP=G and H∩P≦HpG, where HpG is the largest permutable subgroup of G contained in H. A group G is called CPT-group if c-permutability is transitive in G. A number of new characterizations of finite solvable CPT-groups are given.
Let G be a finite group. A subgroup H of G is said to be s-permutable in G if H permutes with all Sylow subgroups of G. Let H be a subgroup of G and let HsG be the subgroup of H generated by all those subgroups of H which are s-permutable in G. A subgroup H of G is called n-embedded in G if G has a normal subgroup T such that HG = HT and H ∩ T ≦ HsG, where HG is the normal closure of H in G. We investigate the influence of n-embedded subgroups of the p-nilpotency and p-supersolvability of G.
A number of authors have studied the structure of a finite group G under the assumption that some subgroups of G are well located in G. We will generalize the notion of s-permutable and s-permutably embedded subgroups and we will obtain new criterions of p-nilpotency and supersolvability of groups. We also generalize some known results.
Let be a class of groups and G a finite group. We call a set Σ of subgroups of G a G-covering subgroup system for
if whenever . Let p be any prime dividing |G| and P a Sylow p-subgroup of G. Then we write Σp to denote the set of subgroups of G which contains at least one supplement to G of each maximal subgroup of P. We prove that the sets Σp and Σp∪Σq, where q≠p, are G-covering subgroup systems for many classes of finite groups.