On Partially Saturated Formations and Residuals of Finite Groups

Shemetkov, L. A.

doi:10.1081/agb-100105992

Cited by 12 publications

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“…The problem of finding of local definitions of ω-saturated formations was considered in [7] and [3]. While solving this problem the following concept of small centralizer was useful (see [8]).…”

Section: Theorem 32 a Non-empty Formation Has A Local Satellite If An...mentioning

confidence: 99%

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Local definitions of formations of finite groups

Shemetkov

2012

J Math Sci

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A problem of constructing of local definitions for formations of finite groups is discussed in the article. The author analyzes relations between local definitions of various types. A new proof of existence of an ω-composition satellite of an ω-solubly saturated formation is obtained. It is proved that if a non-empty formation of finite groups is X-local by Förster, then it has an X-composition satellite.

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Section: Theorem 32 a Non-empty Formation Has A Local Satellite If An...mentioning

confidence: 99%

“…The concept of local definitions of saturated formations was considered for the first time by W. Gaschütz [1]. Following [3], we formulate it in the general form.…”

Section: Introductionmentioning

confidence: 99%

Local definitions of formations of finite groups

Shemetkov

2012

J Math Sci

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“…The origin was the need of studying arbitrary formations by functional methods. The main idea, which was developed in [BBS97] and [She01], is the following:…”

Section: Chapter 1 X-local Formationsmentioning

confidence: 99%

On partially saturated formations of finite groups

Ballester‐Bolinches¹,

Calvo²,

Shemetkov³

2007

Sb. Math.

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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...

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“…Термин "спутник" впервые был введен в работе [16], и его появление было вызвано потребностью изучения произвольных формаций функциональными методами. Сформулируем, следуя [16], [21]…”

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“…Рассуждая по индукции, мы можем считать, что Mминимальная нормальная подгруппа. Из условия леммы непосредственно следует, что все EX-главные Нам понадобятся некоторые обозначения из [21]. Пусть L -некоторый непустой класс простых групп, E cL -класс всех групп, у которых все главные EL-факторы центральны.…”

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О Частично Насыщенных Формациях Конечных Групп

Баллестер-Болинше¹,

Ballester‐Bolinches²,

Кальво³

et al. 2007

Матем. сб.

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О частично насыщенных формациях конечных групп Рассматриваются различные виды частично насыщенных формаций и связи между ними. Установлено, что частично насыщенные формации могут быть охарактеризованы как классы конечных групп с обобщенными центральными рядами. Доказана теорема о разложении F G-модуля в сумму двух подмодулей с определенными свойствами. Библиография: 26 названий.

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On Partially Saturated Formations and Residuals of Finite Groups

Cited by 12 publications

References 9 publications

Local definitions of formations of finite groups

Local definitions of formations of finite groups

On partially saturated formations of finite groups

О Частично Насыщенных Формациях Конечных Групп

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