R. Esteban-Romero

Products of Finite Groups

The study of finite groups factorised as a product of two or more subgroups has become a subject of great interest during the last years with applications not only in group theory, but also in other areas like cryptography and coding theory. It has experienced a big impulse with the introduction of some permutability conditions. The aim of this book is to gather, order, and examine part of this material, including the latest advances made, give some new approach to some topics, and present some new subjects of research in the theory of finite factorised groups.

Sylow permutable subnormal subgroups of finite groups II

Throughout the paper, the word group means finite group. A subgroup Hofa. group G is said to be S-permutable in G if H permutes with every Sylow subgroup of G. By the results of Kegel [9] and Schmid [12], every S-permutable subgroup is subnormal and the set of all S-permutable subgroups is a sublattice of the lattice of all subnormal subgroups of G. We say that a group G is a PST-group when the above lattices coincide, that is, when every subnormal subgroup is S-permutable. It follows that a group G is a PST-group exactly when the S-permutability is a transitive relation. Subclasses of PST-groups are the class of PT-groups or groups in which permutability is transitive and the class of T-groups or groups in which normality is transitive. Soluble P5T-groups were first studied by Agrawal [1] in 1975, and recently by Alejandre, the first author and Pedraza-Aguilera [2], the authors [3], and Beidleman and Heineken [5]. Soluble PT-groups have been investigated by Zacher [13] in 1964, and more recently by Beidleman, Brewster and Robinson [4]. T-groups have been widely studied [6, 8, 10, 11]. The paper [3] provides a unified viewpoint for the classes of soluble PST, PT and Tgroups in terms of their Sylow structure. The approach we have been following started in a paper of Bryce and Cossey [6], where a local version of some of the results on T-groups was established. This approach was also followed in the papers [2, 5]. One of our purposes in this paper is to give a local version of the classical theorems of Gaschiitz, Zacher and Agrawal by using the results of [3]. We also provide new local

On a Class of p-Soluble Groups

Let p be a prime. The class of all p-soluble groups G such that every p-chief factor of G is cyclic and all p-chief factors of G are G-isomorphic is studied in this paper. Some results on T-, PT-, and PST-groups are also obtained.

On minimal non-supersoluble groups

This paper has been published in Revista Matematica Iberoamericana, 23(1):127-142 (2007). Copyright 2007 by Real Sociedad Matematica Espanola and European Mathematical Society Publishing House. The final publication is available at http://rmi.rsme.es http://www.ems-ph.org/journals/journal.php?jrn=rmi http://projecteuclid.org/euclid.rmi/1180728887

On finite minimal non-nilpotent groups

First published in Proceedings of the American Mathematical Society in volume 12, number 6:961-963 (2009), published by the American Mathematical Society http://dx.doi.org/10.1090/S0002-9939-05-07996-7 http://www.ams.org/journals/proc/2005-133-12/S0002-9939-05-07996-7/home.html

A note on finite PST-groups

Abstract A finite group G is said to be a 𝒫𝒮𝒯-group if, for subgroups H and K of G with H Sylow-permutable in K and K Sylow-permutable in G, it is always the case that H is Sylowpermutable in G. A group G is a 𝒯*-group if, for subgroups H and K of G with H normal in K and K normal in G, it is always the case that H is Sylow-permutable in G. In this paper, we show that the classes of finite 𝒫𝒮𝒯-groups and finite 𝒯*-groups coincide. A new characterization of soluble 𝒫𝒮𝒯-groups is also presented.

ON 𝔛-SATURATED FORMATIONS OF FINITE GROUPS

This is an Authors Original Manuscript of an article submitted for consideration in Communications in Algebra 33(4):1053-1064 (2005) (copyright Taylor & Francis). Communications in Algebra is available online at http://www.tandfonline.com/10.1081/AGB-200053809 DOI 10.1081/AGB-200053809 The final publication is available at link.springer.comABSTRACT In the paper, a Frattini-like subgroup associated with a class 𝔛 of simple groups is introduced and analyzed. The corresponding 𝔛-saturated formations are exactly the 𝔛-local ones introduced by Förster. Our techniques are also useful to highlight the properties and behavior of ω-local formations. In fact, extensions and improvements of several results of Shemetkov are natural consequences of our study.

On abnormal maximal subgroups of finite groups

An apparatus 115 for processing a substrate 20, comprises an integrated pumping system 155 having a high operating efficiency, small size, and low vibrational and noise levels. The apparatus 115 comprises a chamber, such as a load-lock chamber 110, transfer chamber 115, or process chamber 120. An integrated pump 165 is abutting or adjacent to one of the chambers 110, 115, 120 for evacuating gas from the chambers. In operation, the pump is located within the actual envelope or footprint of the apparatus and has an inlet 170 connected to a chamber 110, 115, 120, and an outlet 175 that exhausts the gas to atmospheric pressure. Preferably, the integrated pump 165 comprises a pre-vacuum pump or a low vacuum pump and is housed in a noise reducing enclosure having means for moving the pump between locations and means for stacking pumps vertically in use.

A QUESTION FROM THE KOUROVKA NOTEBOOK ON FORMATION PRODUCTS

It is shown in this paper that if X is a class of simple groups such that (X) = charX, the X-saturated formation H generated by a finite group cannot be expressed as the Gaschutz product F G of two nonX-saturated formations if H 6 G. It answers some open questions on products of formations. The relation between !-saturated and Xsaturated formations is also discussed.

ON A QUESTION OF BEIDLEMAN AND ROBINSON

ABSTRACT In [[1], Theorem A], Beidleman and Robinson proved that if a group satisfies the permutizer condition, it is soluble, its chief factors have order a prime number or and induces the full group of automorphisms in the chief factors of order . In this paper, we show that the converse of this theorem is false by showing some counterexamples. We also find some sufficient conditions for a group satisfying the converse of that theorem to satisfy the permutizer condition.