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In the paper, a Frattini-like subgroup associated with a class X of simple groups is introduced and analysed. The corresponding X-saturated formations are exactly the X-local ones introduced by Förster. Our techniques are also very useful to highlight the properties and behaviour of ω-local formations. In fact, extensions and improvements of several results of Shemetkov are natural consequences of our study.

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An easy-to-use procedure is presented for improving theε-constraint method for computing the efficient frontier of the portfolio selection problem endowed with additional cardinality and semicontinuous variable constraints. The proposed method provides not only a numerical plotting of the frontier but also an analytical description of it, including the explicit equations of the arcs of parabola it comprises and the change points between them. This information is useful for performing a sensitivity analysis as well as for providing additional criteria to the investor in order to select an efficient portfolio. Computational results are provided to test the efficiency of the algorithm and to illustrate its applications. The procedure has been implemented in Mathematica.

It is shown in this paper that if X is a class of simple groups such that TT(JE) = char X, the 3E-saturated formation f) generated by a finite group cannot be expressed as the Gaschiitz product J? o

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In this paper criteria for a product of formations to be X-local, X a class of simple groups, are obtained. Some classical results on products of saturated formations appear as particular cases.

A Javi, que ha vivido muy de cerca el desarrollo de esta tesis desde el principio. Y casi hasta el final. Gracias por los chistes. Porúltimo, quiero agradecer a mis amigos los buenos ratos que me han hecho pasar, incluso en losúltimos meses. Gracias a Marta, mi prima y mejor amiga. A Jordi, mi Dr. House, que dice siempre todo lo que piensa. Desde que lo conocí, no me ha faltado su apoyo. Y aÁngel, Víctor, Marcos, Laura, Javi Gascó, Pilar, Roma y Lucy. Un recuerdo en la añoranza para Xiada. Gracias a mis amigos de Würzburg, especialmente a Grhgìrhc, desordenadamente listo, con quien puedo estar toda la vida sin aburrirme. Euqarist¸. 12 CHAPTER 1. X-LOCAL FORMATIONS The class F • G is again a formation and if F is closed under taking subnormal subgroups, then FG = F • G (see [DH92; IV, 1.7 and 1.8]). Definition 1.1.2. A formation F is said to be saturated when G/Φ(G) ∈ F implies that G ∈ F, where Φ(G) denotes the Frattini subgroup of G. Gaschütz [Gas63] introduced the concept of local formation, which enabled him to construct a rich family of saturated formations. Definition 1.1.3. • A formation function f assigns to every p ∈ P a (perhaps empty) formation f (p). • If f is a formation function, then the local formation LF(f) defined by f is the class of all groups G such that if H/K is a chief factor of G, then G/ C G (H/K) ∈ f (p) for all p ∈ π(H/K). • A formation F is said to be local if there exists a formation function f such that F = LF(f). The following remarkable theorem characterises local formations. It was proved by Gaschütz and Lubeseder in the soluble universe and later generalised by Schmid to the general finite one. It is now known as the Gaschütz-Lubeseder-Schmid theorem. Theorem 1.1.4 (Gaschütz-Lubeseder-Schmid [DH92; IV, 4.6]). A formation F is saturated if and only if F is local. Baer followed another approach to extend the theorem of Gaschütz and Lubeseder to the finite universe. He used a different notion of local formation, in which non-abelian chief factors were treated with more flexibility than abelian ones. This led him to find a new family of formations, the Baer-local formations, containing the local ones. Definition 1.1.5. • A Baer function assigns to every simple group J a class of groups f (J) such that f (C p) is a formation for every p ∈ P. • If f is a Baer function, then the Baer-local formation or Baer formation BLF(f) defined by f is the class of all groups G such that if H/K is a chief factor of G, then G/ C G (H/K) ∈ f (J), where J is the composition factor of H/K. 1.1. PRELIMINARY CONCEPTS 13 • A formation F is said to be Baer-local if there exists a Baer function f such that F = BLF(f). Shemetkov introduced in [She75] the concept of composition formation. This notion is equivalent to the one of Baer-local formation. Definition 1.1.6. A formation F is said to be solubly saturated when, for every group G, the condition G/Φ(G S) ∈ F implies that G ∈ F, where G S denotes the soluble radical of G. Baer proved the following theorem: Theorem 1.1.7 (Baer, [DH92; IV, 4.17]). A formatio...