A. Ballester-Bolinches

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.

Sufficient conditions for supersolubility of finite groups

Abstract In this paper sufficient conditions for the supersolubility of finite groups are given under the assumption that the maximal subgroups of Sylow subgroups of the group and the maximal subgroups of Sylow subgroups of the Fitting subgroup are well-situated in the group. That will improve earlier results of Srinivasan [7], Asaad et al. [1] and Ballester-Bolinches [2].

C -Supplemented subgroups of finite groups

A subgroup H of a group G is said to be c-supplemented in G if there exists a subgroup K of G such that HKa G and H\ K is contained in CoreGOHU .W e follow Halls ideas to characterize the structure of the finite groups in which every subgroup is c-supplemented. Properties of c-supplemented subgroups are also applied to determine the structure of some finite groups.

Finite groups with some C-normal minimal subgroups

Abstract Let G be a finite group. The question of how the properties of its minimal subgroups influence the structure of G is of considerable interest for some scholars. Several authors have investigated this question by using normal or quasinormal conditions. In this paper we use c -normal condition on minimal subgroups to characterize the structure of G through the theory of formations.

A note on minimal subgroups of finite groups

1 Introduction and Preliminaries All groups considered in this note will be finite. Recall that a minimal subgroup of a finite group is a subgroup of prime order. Many authors have investigated the structure of a finite group G, under the assumption that all minimal subgroups of G are well-situated in the group. Ito [7;III, 5.3] proved that if G is a group of odd order and all minimal subgroups of G lie in the center of G, then G is nilpotent. An extension of Itos result is the following statement [7;IV,p.435]: If for an odd prime p, every subgroup of G of order p lies in the center of G, then G is p-nilpotent. If all element of G of orders 2 and 4 lie in the center of G, then G is 2-nilpotent. Buckley [4] proved that if G is a group of odd order and all minimal subgroups of G are normal in G, then G is supersoluble. Later Shaalan [8] proved that if G is a finite group and every subgroup of G of prime order or order 4 is π-quasinormal in G, then G is supersoluble. Recall that a subgroup H of a group G is...

ℏ-Normalizers and local definitions of saturated formations of finite groups

We define, in each finite groupG, h-normalizers associated with a Schunck class ℏ of the formEΦ f with f a formation. We use these normalizers in order to give some sufficient conditions for a saturated formation of finite groups to have a maximal local definition.

On finite T-groups

This paper has been published in Journal of the Australian Mathematical Society 75(2):181-192 (2003). The final publication is available at Cambridge University Press Journals, http://journals.cambridge.org/abstract_S1446788700003712 http://dx.doi.org/10.1017/S1446788700003712

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