On partially saturated formations of finite groups.

by Calvo López, Clara

ABSTRACT This thesis deals with finite groups. More precisely, it studies formations, which are classes of groups closed under taking homomorphic images and subdirect products. The concept of X-local formation, where X is a class of simple groups, generalises the concepts of local formation and Baer-local formation. It was introduced by F¨orster with the aim of generalising the well-known theorems of Gasch¨utz-Lubeseder-Schmid and Baer. The first one states that a formation is saturated if and only if it is local. The second one characterises Baer-local formations as the solubly saturated ones. F¨orster introduced a Frattini-like subgroup associated with the class X and which allowed him to introduce the concept of X-saturation. In the first chapter of the thesis a new Frattini-like subgroup is defined. This subgroup allows us to give a more natural generalisation of the above-mentioned theorems. Moreover, X-local formations are characterised as the classes of groups with generalised central chief factors. In the second chapter the relation between this family of formations and the one of !-saturated formations, where ! is a set of primes, is studied. The third chapter is devoted to analyse some problems on the product of two formations. A classical result states that the product of two saturated formations is a saturated formation. This result is not true in general for solubly saturated formations. We give conditions on two X-saturated formations to ensure that the product is an X-saturated formation. As a corollary we obtain a result for products of Baer-local formations. The starting point of chapter four is a question posed by Skiba in the Kourovka Notebook dealing Baer-local formations generated by a group. We study a more general question by using the concept of X-local formation. Moreover, we give a complete description of the factorisations of an X-local formation generated by a group.