On partially saturated formations of finite groups

Abstract

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...

On finite groups with special Sylow p-subgroups

“…A formation F is called psaturated (see [17]) if G/Φ(O p (G)) ∈ F always implies G ∈ F. For example, the class of all finite p-supersoluble groups with an abelian Hall p -subgroup is a p-saturated formation, and it is nonsaturated.…”

Local definitions of formations of finite groups

“…X-local formations were investigated in [20,19,21,22,23]. In [24] it was proved with help of some lemmas in [22] that every X-local formation has a X + -composition satellite. Now we give a direct proof of that fact.…”