Finite groups with some pronormal subgroups

A subgroup H of finite group G is called pronormal in G if for every element x of G, H is conjugate to H ^x in 〈H, H ^x 〉. A finite group G is called PRN-group if every cyclic subgroup of G of prime order or order 4 is pronormal in G. In this paper, we find all PRN-groups and classify minimal non-PRN-groups (non-PRN-group all of whose proper subgroups are PRN-groups). At the end of the paper, we also classify the finite group G, all of whose second maximal subgroups are PRN-groups.     

On the application of some concepts of ring theory to the study of the influence of systems of subgroups of a group

We study groups in which the family L _non-nn(G) of all not nearly normal subgroups has the Krull dimension. A subgroup H of a group G is called nearly normal if H has finite index in its normal closure. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 5, pp. 657–668, May, 2008.     

π-quasinormally embedded and c-supplemented subgroup of finite group

Suppose that H is a subgroup of a finite group G. H is called π-quasinormal in G if it permutes with every Sylow subgroup of G; H is called π-quasinormally embedded in G provided every Sylow subgroup of H is a Sylow subgroup of some π-quasinormal subgroup of G; H is called c-supplemented in G if there exists a subgroup N of G such that G = HN and H ∩ N ⩽ H _G = Core _G (H). In this paper, finite groups G satisfying the condition that some kinds of subgroups of G are either π-quasinormally embedded or c-supplemented in G, are investigated, and theorems which unify some recent results are given.      

The influence of <Emphasis Type="Italic">M</Emphasis>-supplemented subgroups on the structure of finite groups

A subgroup H of a group G is said to be M-supplemented in G if there exists a subgroup B of G such that G = HB and T B < G for every maximal sub-group T of H. Moreover, a subgroup H is called c-supplemented in G if there exists a subgroup K such that G = HK and H ∩ K ≤ H _G where H _G is the largest normal subgroup of G contained in H. In this paper we give some conditions of supersolv-ability of finite group under assumption that some primary subgroups have some kinds of supplements, which are generalizations of some recent results.     

On Nearly S-Permutable Subgroups of Finite Groups

We say that a subgroup H of a finite group G is nearly S-permutable in G if for every prime p such that (p, |H|) = 1 and for every subgroup K of G containing H the normalizer N _K (H) contains some Sylow p-subgroup of K. We study the structure of G under the assumption that some subgroups of G are nearly S-permutable in G.     

On weakly τ-quasinormal subgroups of finite groups

Let G be a finite group and H a subgroup of G. We say that: (1) H is τ-quasinormal in G if H permutes with all Sylow subgroups Q of G such that (|Q|, |H|) = 1 and (|H|, |Q ^G |) ≠ 1; (2) H is weakly τ-quasinormal in G if G has a subnormal subgroup T such that HT = G and T ∩ H ≦ H _τG , where H _τG is the subgroup generated by all those subgroups of H which are τ-quasinormal in G. Our main result here is the following. Let ℱ be a saturated formation containing all supersoluble groups and let X ≦ E be normal subgroups of a group G such that G/E ∈ ℱ. Suppose that every non-cyclic Sylow subgroup P of X has a subgroup D such that 1 < |D| < |P| and every subgroup H of P with order |H| = |D| and every cyclic subgroup of P with order 4 (if |D| = 2 and P is non-Abelian) not having a supersoluble supplement in G is weakly τ-quasinormal in G. If X is either E or F* (E), then G ∈ ℱ.     

On galois groups and their maximal 2-subgroups

LetG be a finite group of even order, having a central element of order 2 which we denote by −1. IfG is a 2-group, letG be a maximal subgroup ofG containing −1, otherwise letG be a 2-Sylow subgroup ofG. LetH=G/{±1} andH=G/{±1}. Suppose there exists a regular extensionL ₁ of ℚ(T) with Galois groupG. LetL be the subfield ofL ₁ fixed byH. We make the hypothesis thatL ₁ admits a quadratic extensionL ₂ which is Galois overL of Galois groupG. IfG is not a 2-group we show thatL ₁ then admits a quadratic extension which is Galois over ℚ(T) of Galois groupG and which can be given explicitly in terms ofL ₂. IfG is a 2-group, we show that there exists an element α ε ℚ(T) such thatL ₁ admits a quadratic extension which is Galois over ℚ(T) of Galois groupG if and only if the cyclic algebra (L/ℚ(T).a) splits. As an application of these results we explicitly construct several 2-groups as Galois groups of regular extensions of ℚ(T).     

Groups with Polycyclic-by-Finite Conjugate Classes of Subgroups

Abstract

Neumann characterized the groups in which every subgroup has finitely many conjugates only as central-by-finite groups. If 𝔛 is a class of groups, a group G is said to have 𝔛-conjugate classes of subgroups if G/Core _G (N _G (H)) ∈ 𝔛 for every subgroup H of G. In this paper, we generalize Neumann's result by showing that a group has polycyclic-by-finite classes of conjugate subgroup if and only if it is central-by-(polycyclic-by-finite).     

On <Emphasis Type="Italic">p</Emphasis>-nilpotency and minimal subgroups of finite groups

We call a subgroup H of a finite group G c-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K ⩽ core(H). In this paper it is proved that a finite group G is p-nilpotent if G is S ₄-free and every minimal subgroup of P ∩ G ^N is c-supplemented in N _G (P), and when p = 2 P is quaternion-free, where p is the smallest prime number dividing the order of G, P a Sylow p-subgroup of G. As some applications of this result, some known results are generalized.     

On Π-Supplemented Subgroups of a Finite Group

A subgroup H of a finite group G is said to satisfy Π-property in G if for every chief factor L/K of G, |G/K: N_G/K(HK/K ∩ L/K)| is a π(HK/K ∩ L/K)-number. A subgroup H of G is called Π-supplemented in G if there exists a subgroup T of G such that G = HT and H ∩ T ≤ I ≤ H, where I satisfies Π-property in G. In this article, we investigate the structure of a finite group G under the assumption that some primary subgroups of G are Π-supplemented in G. The main result we proved improves a large number of earlier results.     

On solubility and supersolubility of some classes of finite groups

Let G be a finite group and H a subgroup of G. Then H is said to be S-permutable in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-permutable in G. Then we say that H is S-embedded in G if G has a normal subgroup T and an S-permutable subgroup C such that T ∩ H HsG and HT = C. Our main result is the following Theorem A. A group G is supersoluble if and only if for every non-cyclic Sylow subgroup P of the generalized Fitting subgrou...     

On totally inert simple groups

A subgroup H of a group G is inert if |H: H ∩ H ^g | is finite for all g ∈ G and a group G is totally inert if every subgroup H of G is inert. We investigate the structure of minimal normal subgroups of totally inert groups and show that infinite locally graded simple groups cannot be totally inert.     

On s-quasinormal and c-normal subgroups of a finite group

Let F be a saturated formation containing the class of supersolvable groups and let G be a finite group. The following theorems are shown： （1） G ∈ F if and only if there is a normal subgroup H such that G/H ∈ F and every maximal subgroup of all Sylow subgroups of H is either c-normal or s-quasinormally embedded in G; （2） G ∈F if and only if there is a soluble normal subgroup H such that G/H∈F and every maximal subgroup of all Sylow subgroups of F（H）, the Fitting subgroup of H, is either e-normally or s-quasinormally embedded in G.     

A Note on c-Supplemented Subgroups of Finite Groups

A subgroup H of a finite group G is said to be c-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K is contained in H _G , where H _G is the largest normal subgroup of G contained in H. In this article, we prove that G is solvable if every subgroup of prime odd order of G is c-supplemented in G. Moreover, we prove that G is solvable if and only if every Sylow subgroup of odd order of G is c-supplemented in G. These results improve and extend the classical results of Hall's articles of (1937) and the recent results of Ballester-Bolinches and Guo's article of (1999), Ballester-Bolinches et al.'s article of (2000), and Asaad and Ramadan's article of (2008).     

On self-normality and abnormality in the alternating groups

It is known that in a finite solvable group G, a subgroup H is abnormal if and only if every subgroup of G containing H is self-normalizing in G. Although, in general, the assumption of solvability cannot be dropped, in this paper we prove the theorem for the special case G = A_n and H a second maximal intransitive subgroup of A_n.Received: 1 July 2003     

Isometry Groups of Proper Hyperbolic Spaces

Let X be a proper hyperbolic geodesic metric space and let G be a closed subgroup of the isometry group Iso(X) of X. We show that if G is not elementary then for every p ∈ (1, ∞) the second continuous bounded cohomology group H²_cb(G, L^p(G)) does not vanish. As an application, we derive some structure results for closed subgroups of Iso(X). Partially supported by Sonderforschungsbereich 611.     

Compactifications and algebraic completions of limit groups

This paper considers the existence of nondiscrete embeddings Γ ↦ G, where Γ is an abstract limit group and G is topological group. Namely, it is shown that a locally compact group G that admits a nondiscrete nonabelian free subgroup F admits a nondiscrete copy of every nonabelian limit group L. In some cases, for instance if the F is of rank 2 and its closure in G is compact or semisimple algebraic, or if L is a surface group (as considered in [6]), L can be chosen with the same closure as F.