Luis M. Ezquerro

• Mathematics
• Catedrático de Álgebra at Public University of Navarre

The influence of the cyclic subgroups of order p or 4 of the focal subgroup of a saturated fusion system over a p-group S is investigated in this paper. Some criteria for normality of S in as well as necessary and sufficient conditions for nilpotency of are given. The resistance of a p-group in which every cyclic subgroup of order p or 4 is normal,...

In this paper we analyse the structure of a finite group of minimal order among the finite non-supersoluble groups possessing a triple factorization by supersoluble subgroups of pairwise relatively prime indices. As an application we obtain some sufficient conditions for a triple factorized group by supersoluble subgroups of pairwise relatively pri...

Given a chief factor H/K of a finite group G, we say that a subgroup A of G avoids H/K if H∩A=K∩A; if HA=KA, then we say that A covers H/K. If A either covers or avoids the chief factors of some given chief series of G, we say that A is a partial CAP-subgroup of G. Assume that G has a Sylow p-subgroup of order exceeding pk. If every subgroup of ord...

The aim of this paper is to obtain a bound for the \(p\) -length of a \(p\) -soluble group \(G\) whose elements of order \(p\) or order \(4\) (if \(p = 2\) ) of a Sylow \(p\) -subgroup of a residual subgroup of \(G\) are contained in the \(k\) -th term of the upper central series of a Sylow \(p\) -subgroup of \(G\) .

Subgroups \(A\) and \(B\) of a finite group are said to be mutually permutable (respectively, M-permutable and \({{\mathrm{sn}}}\) -permutable) if \(A\) permutes with every subgroup (respectively, every maximal subgroup and every subnormal subgroup) of \(B\) and viceversa. If every subgroup of \(A\) permutes with every subgroup of \(B\) , then the...

The main aim of this paper is to give structural information of a finite group of minimal order belonging to a subgroup-closed class of finite groups and whose p-length is greater than 1, p a prime number. Alternative proofs and improvements of recent results about the influence of minimal p-subgroups on the p-nilpotence and p-length of a finite gr...

The main purpose of this paper is to analyze the influence on the structure of a finite group of some subgroups lying in the hypercenter. More precisely, we prove the following: Let \(\mathfrak{F}\) be a Baer-local formation. Given a group G and a normal subgroup E of G, let \(Z_\mathfrak{F} (G)\) contain a p-subgroup A of E which is maximal being...

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.

A subgroup A of a group G is said to be a CAP-subgroup of G if for any chief factor H/K of G, there holds H8745A=K8745A or HA=KA. We investigate the influence of CAP-subgroups on the structure of finite groups. Some recent results are generalized.

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

In this paper we study the influence of the partial cover and avoidance property on the subgroups of some relevant families of subgroups in a finite group.

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 o...

A classical topic of research in Finite Group Theory is the following: What is the influence on the structure of the group of the fact that all members of some relevant family of subgoups enjoy a given embedding property? The cover-avoidance property is a subgroup embedding prop-erty that has recovered much attention in the last few years. In this...

In this paper, we prove the following result. Let \({\mathfrak{F}}\) be a saturated formation and Σ a Hall system of a soluble group G. Let X be a w-solid set of maximal subgroups of G such that Σ reduces into each element of X. Consider in G the following three subgroups: the \({\mathfrak{F}}\)-normalizer D of G associated with Σ; the X-prefrattin...

Many group theorists all over the world have been trying in the last twenty-five years to extend and adapt the magnificent methods of the Theory of Finite Soluble Groups to the more ambitious universe of all finite groups. This is a natural progression after the classification of finite simple groups but the achievements in this area are scattered...

Hypercentrally embedded subgroups of finite groups can be characterized in terms of permutability as those subgroups which permute with all pronormal subgroups of the group. Despite that, in general, hypercentrally embedded subgroups do not permute with the intersection of pronormal subgroups, in this paper we prove that they permute with certain r...

We prove the following results. Let π be a set of prime numbers and G a finite π-soluble group. Consider U,V≤G and H∈Hall π (G) such that H∩V∈Hall π (V) and 1≠H∩U∈Hall π (U). Suppose also that H∩U is a Hall π-subgroup of some S-permutable subgroup of G. Then H∩U∩V∈Hall π (U∩V) and 〈H∩U,H∩V〉∈Hall π (〈U,V〉). Therefore, the set of all S-permutably emb...

We say that a formation F-fraktur sign of finite groups has the Kegel property if F-fraktur sign contains every group of the form G = AB = BC = CA with A, B, C in F-fraktur sign. Vasil'ev asked the following question in the Kourovka Notebook: if F-fraktur sign is a soluble Fitting formation of finite groups with the Kegel property must F-fraktur si...

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, if F is subg...

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 supersolu...

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

Given two subgroups U,V of a finite group which are subnormal subgroups of their join 〈U,V〉 and a formation F, in general it is not true that 〈U,V〉F=〈UF,VF〉. A formation is said to have the Wielandt property if this equality holds universally. A formation with the Wielandt property must be a Fitting class. Wielandt proved that the most usual Fittin...

Following the theory of operators created by Wielandt, we ask for what kind of formations $\mathfrak{F}$ and for what kind of subnormal subgroups $U$ and $V$ of a finite group $G$ we have that the $\mathfrak{F}$-residual of the subgroup generated by two subnormal subgroups of a group is the subgroup generated by the $\mathfrak{F}$-residuals of the...

In this paper we characterise the subgroup–closed Fitting formations of finite groups which are saturated. This is an extension of the Bryce and Cossey result proving the saturation of all subgroup-closed Fitting formations of finite soluble groups.

Introduction. All groups considered in the sequel are finite. Let (£ and 3 denote the formations of groups which consist of collections of groups that respectively either split over each normal subgroup (nC-groups) or for which the groups do not possess nontrivial Frat-tini chief factors [8]. The purpose of this article is to develop and expand a c...

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...

UDC 512.54 All groups considered in this paper are finite. The reader is assumed to be familiar with the theory of saturated formations of finite groups. We shall adhere to the notation used in [5]; this book is the main reference for the basic notation, terminology, and results. First we introduce the question that we aim to analyze in this paper...

All groups considered are finite. In recent years a number of generalizations of the classic Jordan-Hölder Theorem have been obtained (see [7], Theorem A.9.13): in a finite group G a one-to-one correspondence as in the Jordan-Holder Theorem can be defined preserving not only G -isomorphic chief factors but even their property of being Frattini or n...

This paper examines the following question. If $$\mathcal{H}$$ and $$\mathcal{F}$$ are saturated formations then $$\mathcal{H}_\mathcal{F} $$ is defined to be the class of all soluble groups whose $$\mathcal{H} - normalizers$$ belong to $$\mathcal{F}$$ . In general $$\mathcal{H}_\mathcal{F} $$ is a formation, but need not be a saturated formation....

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...

We analyze the influence of the maximal subgroups in the structure of a finite group G by means of the index complex introduced by Deskins in Proc. Sympos. Pure Math. vol. 1, Amer. Math. Soc. 1959, pp. 100-104.

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