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RESEARCH PRODUCT

On partially saturated formations of finite groups.

Clara Calvo López

subject

noneFacultat de Matemàtiques512MathematicsUNESCO::MATEMÁTICAS

description

This thesis deals with finite groups. More precisely, it studies formations,which are classes of groups closed under taking homomorphic images andsubdirect products. The concept of X-local formation, where X is a classof simple groups, generalises the concepts of local formation and Baer-localformation. It was introduced by F¨orster with the aim of generalising thewell-known theorems of Gasch¨utz-Lubeseder-Schmid and Baer. The first onestates that a formation is saturated if and only if it is local. The second onecharacterises Baer-local formations as the solubly saturated ones. F¨orsterintroduced a Frattini-like subgroup associated with the class X and whichallowed him to introduce the concept of X-saturation. In the first chapterof the thesis a new Frattini-like subgroup is defined. This subgroup allowsus to give a more natural generalisation of the above-mentioned theorems.Moreover, X-local formations are characterised as the classes of groups withgeneralised central chief factors. In the second chapter the relation betweenthis family of formations and the one of !-saturated formations, where ! isa set of primes, is studied. The third chapter is devoted to analyse someproblems on the product of two formations. A classical result states that theproduct of two saturated formations is a saturated formation. This resultis not true in general for solubly saturated formations. We give conditionson two X-saturated formations to ensure that the product is an X-saturatedformation. As a corollary we obtain a result for products of Baer-localformations. The starting point of chapter four is a question posed by Skibain the Kourovka Notebook dealing Baer-local formations generated by agroup. We study a more general question by using the concept of X-localformation. Moreover, we give a complete description of the factorisations ofan X-local formation generated by a group.

http://hdl.handle.net/10550/15783