A characterization of the hypercyclically embedded subgroups of finite groups

A normal subgroup H of a finite group G is said to be hypercyclically embedded in G if every chief factor of G below H is cyclic. The major aim of the present paper is to characterize the normal hypercyclically embedded subgroups E of a group G by means of the embedding of the maximal and minimal subgroups of the Sylow subgroups of the generalized Fitting subgroup of E.     

On second maximal subgroups of Sylow subgroups of finite groups

Finite groups in which the second maximal subgroups of the Sylow p-subgroups, p a fixed prime, cover or avoid the chief factors of some of its chief series are completely classified.     

Mutually permutable subgroups and Fitting classes

Let G = AB be the mutually permutable product of the subgroups A and B. It is shown that if A and B are contained in a Fitting class , then the commutator subgroup G′ of G is also contained in . Received: 14 August 2006 Revised: 17 September 2006     

Mutually permutable subgroups and group classes

We consider the product AB of two finite mutually permutable subgroups A, B and find some subnormal subgroups of the product. This leads to local and otherwise generalized statements about products of supersolvable groups.Received: 19 May 2004     

On the normal indices of proper subgroups of finite groups

We extended the normal index from maximal subgroups to proper subgroups. We give a quantitative version of all results obtained by using c-normal subgroups and obtain some new characterizations of solvable, supersolvable and nilpotent groups by the normal indices of proper subgroups.     

The influence of minimal subgroups of focal subgroups on the structure 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 such that G=HK and H∩K=1. In this paper, it is proved that a finite group G is p-nilpotent provided p is the smallest prime number dividing the order of G and every minimal subgroup of the p-focal subgroup of G is complemented in N_G(P), where P is a Sylow p-subgroup of G. As some applications, some interesting results related with complemented minimal subgroups of focal subgroups are obtained.     

On semi cover-avoiding subgroups of finite groups

A subgroup H of a finite group G is said to have the semi cover-avoiding property in G if there is a normal series of G such that H covers or avoids every normal factor of the series. In this paper, some new results are obtained based on the assumption that some subgroups have the semi cover-avoiding property in the group.     

The existence of large commutator subgroups in factors and subgroups of non-nilpotent groups

We prove that any finite non-nilpotent group G must possess certain factors K / M with a large commutator subgroup, where M is nilpotent. These results are related to recent results by M. Herzog and the authors on large commutator subgroups.Received: 26 August 2004     

On c-normality of certain subgroups of prime power order of finite groups

In this paper, we study the structure of the finite group G given that certain subgroups of prime power order are well-situated, which means that they are normally complemented modulo their normal core.Received: 14 October 2004; revised: 12 January 2005     

The influence of $\pi$-quasinormality of some subgroups of a finite group

A subgroup H of G is said to be $\pi$-quasinormal in G if it permute with every Sylow subgroup of G. In this paper, we extend the study on the structure of a finite group under the assumption that some subgroups of G are $\pi$-quasinormal in G. The main result we proved in this paper is the following:Theorem 3.4. Let ${\cal F}$ be a saturated formation containing the supersolvable groups. Suppose that G is a group with a normal subgroup H such that $G/H \in {\cal F}$, and all maximal subgroups of any Sylow subgroup of $F^{*}(H)$ are $\pi$-quasinormal in G, then $G \in {\cal F}$. Received: 10 May 2002     

Finite p-groups all of whose non-abelian proper subgroups are metacyclic

In this paper, we classify the finite p-groups all of whose non-abelian proper subgroups are metacyclic and answer a question posed by Berkovich. Received: 22 June 2005     

Groups with finitely many derived subgroups of non-normal subgroups

A group is called metahamiltonian if all its non-abelian subgroups are normal; it is known that locally soluble metahamiltonian groups have finite derived subgroup. This result is generalized here, by proving that every locally graded group with finitely many derived subgroups of non-normal subgroups has finite derived subgroup. Moreover, locally graded groups having only finitely many derived subgroups of infinite non-normal subgroups are completely described. Received: 25 April 2005     

Finite 2-groups all of whose nonmetacyclic subgroups are generated by involutions

We classify here nonmetacyclic finite 2-groups all of whose nonmetacyclic subgroups are generated by involutions (Theorem 1.1). This solves problem Nr. 710 for p = 2 stated by Y. Berkovich in [1]. For p > 2 the corresponding problem is open. Received: 14 March 2007 Revised: 15 April 2007     

On finite groups with p-decomposable cofactors of subgroups

It is proved that, for a finite group with p-decomposable cofactors of the maximal subgroups, the quotient group by the Fitting subgroup is p-decomposable. This implies that every group all of whose maximal subgroups have nilpotent cofactors is metanilpotent.     

Products of two groups containing cyclic subgroups of index at most 2

It is proved that every group of the form G = AB with subgroups A and B each of which has a cyclic subgroup of index at most 2 is metacyclicby-finite. Received: 13 July 2007     

Direct sum decomposition of geometries based on maximal subgroups

In 1961 J. Tits described a way to define a geometry from a group and a collection of subgroups. Such incidence geometries are now studied by the team of F. Buekenhout in Brussels. Here we present theorems about decomposition of PRI geometries into direct sums and we find the full direct sum decomposition of PRI geometries on solvable groups.     

Derived length of finite groups with restrictions on Sylow subgroups

The dependence of the derived length of a finite solvable group on the orders of nonbicyclic Sylow subgroups of the Fitting subgroup is established.     

Finite groups in which all p-subnormal subgroups form a lattice

O. Kegel, in 1962, introduced the concept of p-subnormal subgroups of a finite group as the subgroups whose intersections with all Sylow p-subgroups of the group are Sylow p-subgroups of the subgroup. The set of p-subnormal subgroup of a finite group is not a lattice in general. In this paper, the class of all finite groups in which all p-subnormal subgroups from a lattice is determined. This is the class of all finite p-soluble groups whose p-length and p′-length, both, are less or equal to 1. The join-semilattice case and the meet-semilattice case are analyzed separately. The authors are supported by Proyecto PB 94-1048 of DGICYT, Ministerio de Educación y Ciencia of Spain.     

Problems related to the Lockett Conjecture on Fitting classes of finite groups

The existence of a solvable non-normal Fitting class F which is not a Lockett class but for which the Lockett Conjecture still holds are studied. We also prove that there exists an ω-local Fitting class F which does not satisfy the Lockett conjecture but the Lockett conjecture still holds under a given condition. As a consequence of our result, a generalized version of the Lausch's problem in the well-known Kourovka Notebook is answered.     

Free subgroups of branch groups

A procedure is described for constructing branch groups on the binary tree, which yields in particular finitely generated branch groups with non-cyclic free subgroups.