Luis M. Ezquerro

SUBGROUPS OF FINITE GROUPS WITH A STRONG COVER-AVOIDANCE PROPERTY

A subgroup A of a group G has the strong cover-avoidance property in G , or A is a strong CAP-subgroup of G , if A either covers or avoids every chief factor of every subgroup of G containing A . The main aim of the present paper is to analyse the impact of the strong cover and avoidance property of the members of some relevant families of subgroups on the structure of a group.

Some permutability properties related to F-hypercentrally embedded subgroups of finite groups☆

Let F denote a saturated formation. In this paper we study some properties of F-hypercentrally embedded subgroups, i.e., those subgroups T of a finite group G such that every chief factor of G between its core and its normal closure is F-central in G. We prove that these subgroups form a sublattice of the lattice of all subgroups of G ,i fF is subgroup-closed. The main result of the paper is the following: if F contains the class of nilpotent groups and G is a soluble group, a subgroup T which permutes with all Sylow subgroups of G is F-hypercentrally embedded in G if and only if T permutes with some F-normalizer of G.  2003 Elsevier Science (USA). All rights reserved.

On Mutually M-Permutable Products of Finite Groups

Abstract In this paper we deduce some structural properties of a finite group which is a mutually M-permutable product of some subgroups with these properties. We prove the solubility of a finite group G which is a mutually M-permutable product of two soluble subgroups. If moreover the factors are supersoluble subgroups of coprime order, then G is supersoluble. Finally, we characterize what kind of linear orderings ≺ , on the set ℙ of all primes, are such that the mutually M-permutable product of subgroups with Sylow tower of type ≺ is a group with a Sylow tower of the same type.

F-Stability of finite groups

The following theorem is proved: “If G is a finite group and F is a saturated formation of solvable groups such that 2 ∉ charF, then every section of G is F-stable if and only if no section of G is isomorphic to any group SA(2,p), p ϵ char F.” This theorem is a generalization of a well-known Glaubermans theorem.

ON CONJUGACY OF SUPPLEMENTS OF NORMAL SUBGROUPS OF FINITE GROUPS

The objective of this paper is to find some sufficient conditions to ensure the conjugacy of supplements of a normal subgroup of a soluble group. 10.1017/S0004972713000506

On formations with the Kegel property

Abstract We say that a formation ℱ of finite groups has the Kegel property if ℱ contains every group of the form G = AB = BC = CA with A, B, C in ℱ. Vasil’ev asked the following question in the Kourovka Notebook: if ℱ is a soluble Fitting formation of finite groups with the Kegel property must ℱ be a saturated formation? We obtain an affirmative answer in the soluble universe in the case when ℱ has the following additional property: for every prime p ∈ char ℱ and every primitive ℱ-group G whose socle is a p -group, lies in ℱ for all primes q ≠ p such that q divides |G | Soc(G )|.

A Characterization of the Class of Finite Groups with Nilpotent Derived Subgroup

The class of all finite groups with nilpotent commutator subgroup is characterized as the largest subgroup-closed saturated formation 𝔉 for which the 𝔉-residual of a group generated by two 𝔉-subnormal subgroups is the subgroup generated by their 𝔉–residuals.

A NOTE ON FINITE GROUPS GENERATED BY THEIR SUBNORMAL SUBGROUPS

Following the theory of operators created by Wielandt, we ask for what kind of formations F and for what kind of subnormal subgroups U and V of a nite group G we have that the F-residual of the subgroup generated by two subnormal subgroups of a group is the subgroup generated by the F-residuals of the subgroups. In this paper we provide an answer whenever U is quasinilpotent and F is either a Fitting formation or a saturated formation closed for quasinilpotent subnormal subgroups.

Correction to "On the Deskins Index Complex of a Maximal Subgroup of a Finite Group"

This note is to correct a mistake in [1]. So, the notation, definitions, and references are those of that paper. If M is maximal subgroup of a finite group G and H/K is a chief factor of G supplemented by M we cannot say in general that H E I (M). For example, if G is the dihedral group of order 30, M a maximal subgroup of G isomorphic to Sym(3), and H = Soc(G), we have that H f I(M) since k(H) = H. (Compare this with the last paragraph of page 236 in [5].) This motivates that Proposition 1 in our paper does not hold. Changing that by Proposition 1* below, we prove below that the five theorems of the paper remain true.

A question in the theory of saturated formations of finite soluble groups

AbstractThis paper examines the following question. If