Hall operators on the set of formations of finite groups

Let π be a nonempty set of primes and let F be a saturated formation of all finite soluble π-groups. It is constructed the saturated formation consisting of all finite π-soluble groups whose F-projectors contain a Hall π-subgroup.

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Date:	2010
Main Authors:	Mekhovich, A.P., Vorob’ev, N.N., Vorob’ev, N.T.
Format:	Article
Language:	English
Published:	Інститут прикладної математики і механіки НАН України 2010
Series:	Algebra and Discrete Mathematics
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/154492
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Cite this:	Hall operators on the set of formations of finite groups / A.P. Mekhovich, N.N. Vorob’ev, N.T. Vorob’ev // Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 1. — С. 72–78. — Бібліогр.: 19 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1544922025-02-09T16:49:26Z Hall operators on the set of formations of finite groups Mekhovich, A.P. Vorob’ev, N.N. Vorob’ev, N.T. Let π be a nonempty set of primes and let F be a saturated formation of all finite soluble π-groups. It is constructed the saturated formation consisting of all finite π-soluble groups whose F-projectors contain a Hall π-subgroup. Research of the second author is partially supported by Belarussian Republic Foun-dation of Fundamental Researches (BRFFI, grant F08M-118) 2010 Article Hall operators on the set of formations of finite groups / A.P. Mekhovich, N.N. Vorob’ev, N.T. Vorob’ev // Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 1. — С. 72–78. — Бібліогр.: 19 назв. — англ. 1726-3255 2001 Mathematics Subject Classification:20D10 https://nasplib.isofts.kiev.ua/handle/123456789/154492 en Algebra and Discrete Mathematics application/pdf Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Let π be a nonempty set of primes and let F be a saturated formation of all finite soluble π-groups. It is constructed the saturated formation consisting of all finite π-soluble groups whose F-projectors contain a Hall π-subgroup.
format	Article
author	Mekhovich, A.P. Vorob’ev, N.N. Vorob’ev, N.T.
spellingShingle	Mekhovich, A.P. Vorob’ev, N.N. Vorob’ev, N.T. Hall operators on the set of formations of finite groups Algebra and Discrete Mathematics
author_facet	Mekhovich, A.P. Vorob’ev, N.N. Vorob’ev, N.T.
author_sort	Mekhovich, A.P.
title	Hall operators on the set of formations of finite groups
title_short	Hall operators on the set of formations of finite groups
title_full	Hall operators on the set of formations of finite groups
title_fullStr	Hall operators on the set of formations of finite groups
title_full_unstemmed	Hall operators on the set of formations of finite groups
title_sort	hall operators on the set of formations of finite groups
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2010
url	https://nasplib.isofts.kiev.ua/handle/123456789/154492
citation_txt	Hall operators on the set of formations of finite groups / A.P. Mekhovich, N.N. Vorob’ev, N.T. Vorob’ev // Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 1. — С. 72–78. — Бібліогр.: 19 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT mekhovichap halloperatorsonthesetofformationsoffinitegroups AT vorobevnn halloperatorsonthesetofformationsoffinitegroups AT vorobevnt halloperatorsonthesetofformationsoffinitegroups
first_indexed	2025-11-28T02:27:44Z
last_indexed	2025-11-28T02:27:44Z
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fulltext	Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 9 (2010). Number 1. pp. 72 – 78 c© Journal “Algebra and Discrete Mathematics” Hall operators on the set of formations of finite groups Andrei P. Mekhovich, Nikolay N. Vorob’ev and Nikolay T. Vorob’ev Communicated by L. A. Shemetkov Dedicated to Professor I.Ya. Subbotin on the occasion of his 60-th birthday Abstract. Let π be a nonempty set of primes and let F be a saturated formation of all finite soluble π-groups. It is constructed the saturated formation consisting of all finite π-soluble groups whose F-projectors contain a Hall π-subgroup. Introduction In the theory of soluble Fitting classes P. Lockett and P. Hauck considered the classes Lπ(F) and Kπ(F). Definition 1 ([1, 2]). Let π be a set of primes and let F be a Fitting class of finite soluble groups. Then Lπ(F) = (G ∈ S : an F-injector of G contains a Hall π-subgroup of G); Kπ(F) = (G ∈ S : a Hall π-subgroup of G belongs to F). In [1] (see also [3, IX, 1.22]) Lockett used the class Lπ(F) to obtain a description of the injectors for a Fitting class product FG. It was proved that Lπ(F) and Kπ(F) are Fitting classes. Furthermore, Kπ(F) = Research of the second author is partially supported by Belarussian Republic Foun- dation of Fundamental Researches (BRFFI, grant F08M-118) 2001 Mathematics Subject Classification: 20D10. Key words and phrases: Hall π-subgroup, π-soluble group, formation of finite groups, saturated formation, canonical satellite, F-projector. Jo u rn al A lg eb ra D is cr et e M at h .A. P. Mekhovich, N. N. Vorob’ev, N. T. Vorob’ev 73 Lπ(F ∩Sπ). Hence we may consider Lπ and Kπ as operators on the set of all Fitting classes for every π. The class Kπ(F) was introduced by Hauck [2] and has been studied in detail by Brison [4] and Cusack [5]. Moreover, Brison [6, 7] applied Kπ(F) to obtain a description of Hall subgroups radicals. Analogously one may consider the following operators on the set of all soluble formations. Definition 2 ([8, 9]). Let π be a set of primes and let F be a formation of finite soluble groups. Then Lπ(F) = (G ∈ S : an F-projector of G contains a Hall π-subgroup of G); Kπ(F) = (G ∈ S : a Hall π-subgroup of G belongs to F). In [9] Blessenohl proved that if F is a saturated formation, then Kπ(F) is a saturated formation. Further L.A. Shemetkov posed the following question in this trend. Problem (see [10, Problem 19]). Let F be a saturated formation of finite groups, Cπ(F) be the class of all groups G such that there exist Hall π- subgroups of G in F and any two of them are conjugate. Is the class Cπ(F) a saturated formation? The positive answer of Problem 19 was given by L.M. Slepova [11] in the class of all π-separable groups for some restrictions to F; in [12] it was shown by E.P. Vdovin, D.O. Revin and L.A. Shemetkov that Cπ(F) is solubly saturated formation for any solubly saturated formation F. However L.A. Shemetkov and A.F. Vasil’ev [13] proved that in general the class Cπ(N) is not a saturated formation, where N is the class of all nilpotent groups. Wenbin Guo and Baojin Li [14] proved that Kπ(F) is a local Fit- ting class for every local Fitting class F. In general N.T. Vorob’ev and V.N. Zagurskii [15] gave the positive answer of Shemetkov’s Problem for soluble ω-local Fitting classes. K. Doerk and T. Hawkes investigated an analog of Problem 19 for the class Lπ(F). It was proved, that if F is a solubly saturated formation, then Lπ(F) is a saturated formation (see [8, Bemerkung]). Note that the analog of the above-mentioned problem has the negative answer for soluble Schunck classes (see [8, Beispiel 1]) and soluble Fitting classes (see [3, IX, 3.15]). A purpose of this paper is to investigate an analog of Shemetkov’s Problem for the class Lπ(F), where F is the saturated formation of all soluble π-groups. Jo u rn al A lg eb ra D is cr et e M at h .74 Hall operators... All groups considered are finite and π-soluble for some fixed nonempty set of primes π. All unexplained notations and terminologies are stan- dard. The reader is refereed to [16], [10] and [3] if necessary. 1. Preliminaries Recall notation and some definitions used in this paper. A group class closed under taking homomorphic images and finite subdirect products is called a formation. A group G is said to be π-soluble if every chief factor of G is either a p-group for some p ∈ π or a π′-subgroup. The complementary set of primes, P\π, is denoted by π′. σ(G) de- notes the set of all distinct prime divisors of the order of a group G. Functions of the form f : P → {formations of groups} are called local satellites (see [10]). For every local satellite f it is defined the class LF (f) = (G : G has f -central chief series), i.e., for every chief factor H/K of G we have G/CG(H/K) ∈ f(p) for every p ∈ π(H/K). If F is a formation such that F = LF (f) for a local satellite f , then the formation F is said to be saturated and f is a local satellite of F. If F is a saturated formation, by [3, IV, 4.3] we have Char(F) = σ(F), where σ(F) = ⋃ {σ(G) : G ∈ F}. A satellite F of a formation F is called canonical if F (p) ⊆ F, and F (p) = NpF (p) for all p ∈ P [17]. Let F be a formation. A subgroup H of a group G is called F-maximal in G provided that (1) H ∈ F, and (2) if H 6 V 6 G and V ∈ F, then H = V . A subgroup H of G is called an F-projector of G if HN/N is F- maximal in G/N for all N EG. By ProjFG we denote the (possibly empty) set of all F-projectors of G. Let F be a saturated formation and let H be a formation. Following [3, IV, 1.1] we denote the class (F ։ H) as follows: (F ։ H) = (G : ProjFG ⊆ H). Jo u rn al A lg eb ra D is cr et e M at h .A. P. Mekhovich, N. N. Vorob’ev, N. T. Vorob’ev 75 If H = ∅, then (F ։ H) = ∅. If RB ⊇ A, then it is said that A/B covered by R. The symbols Gπ, Sπ, Eπ′ , Eπ and Np denote, respectively, a Hall π-subgroup of a group G, the class of all π-soluble groups, the class of all π′-groups, the class of all π-groups and the class of all p-groups. We need some lemmas to prove the main result. Lemma 1 ([18, Lemma 1.2, Lemma 1.3]). Let F = LF (F ) be the forma- tion of all soluble π-groups. Then the following statements hold: (1) F = LF (m), where m(p) = (F ։ F (p)) for all p ∈ P. (2) If V is an F-projector of a group G, then: (a) V covers every m-central chief factor of G. (b) Every chief factor of G covered of the subgroup V is m-central. Lemma 2 ([10, Theorem 15.7]). Let F be a saturated formation and G be a group having σ(F)-soluble F-residual. Then G has F-projectors and any two of them are conjugate. 2. The proof of Theorem First we prove Lemma 3. Let F be a saturated formation of all soluble π-groups. Then the following statements hold: (1) The class Lπ(F) is a formation. (2) Eπ′Lπ(F) = Lπ(F). Proof. (1) If π = ∅, then L∅(F) = Sπ; if π = P, then LP(F) = F. We have saturated formations Sπ and F, and hence the result. Now suppose ∅ ⊂ π ⊂ P. Since a formation F is saturated, by [3, IV, 4.3] we have Char(F) = σ(F). Since σ(F) ⊆ π, a π-soluble group G is σ(F)-soluble. Consequently, the subgroup GF of G is σ(F)-soluble. Let G ∈ Lπ(F), let K ⊳ G and let F be an F-projector of G. Then there exists a Hall π-subgroup Gπ of G such that Gπ ⊆ F . By [10, Lemma 15.2] and [10, Lemma 15.1], we see that GπK/K is a Hall π-subgroup of G/K and FK/K is an F-projector of G/K. Therefore G/K ∈ Lπ(F). Let K1 and K2 be normal subgroups of G such that K1∩K2 = 1 and let G/K1 ∈ Lπ(F) and G/K2 ∈ Lπ(F). Then GπK1/K1 ⊆ FK1/K1 and GπK2/K2 ⊆ FK2/K2, where GπK1/K1 is a Hall π-subgroup of G/K1 Jo u rn al A lg eb ra D is cr et e M at h .76 Hall operators... and GπK2/K2 is a Hall π-subgroup of G/K2, FK1/K1 is an F-projector of G/K1 and FK2/K2 is an F-projector of G/K2. Therefore GπK1 ⊆ FK1 and GπK2 ⊆ FK2. Hence GπK1 ∩GπK2 ⊆ FK1∩FK2. By [18, Lemma 1.4] and [10, Theorem 15.2] we have Gπ(K1∩ K2) ⊆ F (K1 ∩K2), i.e., Gπ ⊆ F . Thus G ∈ Lπ(F). This proves (1). (2) Inclusion Lπ(F) ⊆ Eπ′Lπ(F) is obvious. We show that Eπ′Lπ(F) ⊆ Lπ(F). Let G ∈ Eπ′Lπ(F). Then GLπ(F) ∈ Eπ′ and G/GLπ(F) ∈ Lπ(F). Let Gπ be a Hall π-subgroup of G and let F be an F-projector of G. By [10, Lemma 15.2] and [10, Lemma 15.1], we see, GπG Lπ(F)/GLπ(F) is a Hall π-subgroup of G/GLπ(F) and FGLπ(F)/GLπ(F) is an F-projector of G/GLπ(F). Therefore GπG Lπ(F)/GLπ(F) ⊆ F xGLπ(F)/GLπ(F). By [10, Lemma 15.1], F xGLπ(F)/GLπ(F) is an F-projector of G/GLπ(F), where x ∈ G/GLπ(F). Consequently, \|G/GLπ(F) : F xGLπ(F)/GLπ(F)\| = \|G\| \|F xGLπ(F)\| = \|G\|\|F ∩GLπ(F)\| \|F \|\|GLπ(F)\| = \|G\| \|F \|\|GLπ(F)\| is a π′-number. Since \|GLπ(F)\| is a π′-number, \|G : F \| is a π′-number. Thus a Hall π-subgroup Gπ of G is contained in the F-projector F of G. Hence G ∈ Lπ(F). The lemma is proved. The following theorem shows that if F is a saturated formation, then the formation Lπ(F) is saturated. Theorem. Let F = LF (F ) be the formation of all soluble π-groups. Then Lπ(F) = LF (f) for a local satellite f such that f(p) = { (F ։ F (p)), if p ∈ π, Sπ, if p ∈ π′. Proof. If π = ∅, then L∅(F) = Sπ; if π = P, then LP(F) = F. We have saturated formations Sπ and F, and hence the result. Now suppose ∅ ⊂ π ⊂ P. Since a formation F is saturated, by [3, IV, 4.3] we have Char(F) = σ(F). So a π-soluble group G is σ(F)-soluble. Consequently, the subgroup GF of G is σ(F)-soluble. By Lemma 1 we have F = LF (m), where m is a local satellite of F such that m(p) = F ։ F (p) for all p ∈ P. Jo u rn al A lg eb ra D is cr et e M at h .A. P. Mekhovich, N. N. Vorob’ev, N. T. Vorob’ev 77 We show LF (f) ⊆ Lπ(F). Suppose LF (f) * Lπ(F). Let G be a group of minimal order in LF (f)\Lπ(F). Then G is a monolithic group and K = GLπ(F) is the socle of G. We have \|G/K\| < \|G\|, so by induction, G/K ∈ Lπ(F). If T is an F-projector of G and Gπ is a Hall π-subgroup of G, then by the definition Lπ(F), we have GπK/K ⊆ TK/K. Hence GπK ⊆ TK. Since G is π-soluble, K is either a p-group, where p ∈ π or a normal π′-subgroup. Let K be a p-group, where p ∈ π. Since G ∈ LF (f), G/CG(K) ∈ f(p) = (F ։ F (p)). By Lemma 1, an F-projector T covers K, i.e., K ⊆ T . Therefore T = TK ⊇ GπK ⊇ Gπ. It follows that G ∈ Lπ(F), a contradiction. Now let K ∈ Eπ′ . Lemma 3 implies G ∈ Eπ′Lπ(F) = Lπ(F), a contradiction. We prove the converse inclusion, i.e., Lπ(F) ⊆ LF (f). Suppose Lπ(F) * LF (f). Let H be a group of minimal order in Lπ(F)\LF (f). Then H is a monolithic group and R = HLF (f) is the socle of H. Since H is π-soluble, R is either a p-group, where p ∈ π or a normal π′-subgroup. Let R be a π′-subgroup. By induction, H/R ∈ LF (f). Consequently, all factors of the chief series H ⊃ . . . ⊃ R are f -central. By assumption, H/CH(R) ∈ Sπ = f(p). Hence H ∈ LF (f), a contradiction. Now let R be a p-group, where p ∈ π. If Hπ is a Hall π-subgroup of H and V is an F-projector of H, then by Chunihin’s Theorem [19], we have R ⊆ Hπ. Since H ∈ Lπ(F), Hπ ⊆ V . Consequently, R ⊆ V , i.e., V covers R. Lemma 1 implies that R is m-central chief factor of H. By induction, H/R ∈ LF (f). Consequently, H ∈ LF (f). This final contradiction completes the proof. References [1] P. Lockett, On the theory of Fitting classes of finite soluble groups, Math. Z. 131 (1973) pp. 103–115. [2] P. Hauck, Eine Bemerkung zur kleinsten normalen Fittingklasse, J. Algebra 53 (1978) pp. 395–401. [3] K. Doerk and T. Hawkes, Finite soluble groups, Walter de Gruyter, Berlin-New York, 1992. [4] O.J. Brison, Hall operators for Fitting classes, Arch. Math.[Basel] 80 (33) (1979/80) pp. 1–9. [5] E. Cusack, Strong containment of Fitting classes, J. Algebra 64 (3) (1980) pp. 414–429. [6] O.J. Brison, A criterion for the Hall-closure of Fitting classes, Bull. Austral. Math. Soc. 23 (1981) pp. 361–365. Jo u rn al A lg eb ra D is cr et e M at h .78 Hall operators... [7] O.J. Brison, Hall-closure and products of Fitting classes, J. Austral. Math. Soc. Ser. A. 32 (1984) pp. 145–164. [8] K. Doerk and T. Hawkes, Ein Beispiel aus der Theorie der Schunckklassen, Arch. Math. [Basel] 31 (1978) pp. 539–544. [9] D. Blessenohl, Über Formationen und Halluntergruppen endlicher auflösbarer Gruppen, Math. Z. 142 (3) (1975) pp. 299–300. [10] L.A. Shemetkov, Formations of finite groups, Nauka, Main Editorial Board for Physical and Mathematical Literature, Moscow, 1978 (in Russian). [11] L.M. Slepova, On formations of E F-groups, Doklady AN BSSR 21 (7) (1977) pp. 587–589 (in Russian). [12] E.P. Vdovin, D.O. Revin and L.A. Shemetkov, Formations of finite Cπ-groups, The International Scientific Conference "X The Belarusian Mathematical Conference" November 3–7, 2008, Minsk, Institute of Mathematics of National Academy of Sciences of Belarus (2008) pp. 12–13. [13] L.A. Shemetkov and A.F. Vasil’ev, Nonlocal formations of finite groups, Doklady AN Belarusi 39 (4) (1995) pp. 5–8 (in Russian). [14] Guo Wenbin and Li Baojun, On Shemetkov problem for Fitting classes, Beitr. Algebra und Geom. 48 (1) (2007) pp. 281–289. [15] N.T. Vorob’ev and V.N. Zagurskii, Fitting classes with given properties of Hall subgroups, Matematicheskiye Zametki 78 (2) (2005) pp. 234–240 (in Russian); translated in Mathematical Notes 78 (2) (2005) pp. 213–218. [16] B. Huppert, Endliche Gruppen, Springer–Verlag, Berlin–Heidelberg–New York, 1967. [17] A.N. Skiba and L.A. Shemetkov, Multiply ω-local formations and Fitting classes of finite groups, Matematicheskiye Trudy 2 (1) (1999) pp. 114–147 (in Russian); translated in Siberian Adv. Math. 10 (2) (2000) pp. 112–141. [18] N.T. Vorob’ev, Maximal screens of local formations, Algebra i Logika 18 (2) (1979) pp. 137–161 (in Russian). [19] S.A. Chunihin, Subgroups of finite groups, Nauka i tekhnika, Minsk, 1964 (in Russian). Contact information A. P. Mekhovich Polotsk State Agricultural Economic Col- lege, Oktyabrskaya street, 55, Polotsk, 211413, Belarus E-Mail: amekhovich@yandex.ru N. N. Vorob’ev, N. T. Vorob’ev Vitebsk State University of P.M. Masherov, Moscow Avenue, 33, Vitebsk, 210038, Be- larus E-Mail: vornic2001@yahoo.com Received by the editors: 08.11.2009 and in final form 21.05.2010.

Hall operators on the set of formations of finite groups

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