It is proved that the lattice of all ω-local formations is a complete sublattice of the lattice of all ω-composition formations of finite groups

Nikolay N. Vorob’ev

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Advances in Group Theory and Applicationsc© 2018 AGTA - www.advgrouptheory.com/journal6 (2018), pp. 89–100 ISSN: 2499-1287DOI: 10.32037/agta-2018-006On Sublattices of the Lattice of allω-Composition Formations of Finite Groups *Nikolay N. Vorob’ev(Received Feb. 8, 2018; Accepted Sept. 10, 2018 — Communicated by A. Skiba)AbstractIt is proved that the lattice of all ω-local formations is a complete sublattice of thelattice of all ω-composition formations of finite groups.Mathematics Subject Classification (2010): 20D10, 20F17Keywords: finite group; formation of groups; ω-local formation; ω-composi-tion formation; complete lattice of formations1 IntroductionThroughout this paper, all groups are finite. Moreover, we will use ω todenote a non-empty set of primes and ω ′ = P \ω.Recall that a variety of groups may be defined as a non-empty class ofgroups closed under taking homomorphic images and subcartesian products(see Chap. 1, Sec. 5, Remark 15.53 of [13]). In the universe of all finitegroups the definition of a variety leads us to the concept of the formation: aclass of finite groups F which is closed under taking homomorphic imagesand finite subdirect products is called a formation (Gaschütz [6]). It is wellknown that the lattice of all varieties of groups is complete, modular but* Research of the author supported by the Belarusian Republic Foundation for Fun-damental Research (grant F18U-007) and by State Program of Fundamental Re-search of Republic Belarus (Grant 20160350)90 Nikolay N. Vorob’evis not distributive [13]. Moreover, Skiba proved (see p.91 of [21] or p.137of [26]) that the lattice of all varieties of locally finite groups is a sublatticeof the lattice of all hereditary (in the sence of Mal’cev; see [12]) formations.This circumstance confirms the importance of studying of the lattices offormations.The most useful for applications of the formation theory (in particu-lar, in the theory of formal languages; see [2],[3],[4]) and in the theory oflattices of group classes (see [1],[5],[7],[8],[9],[10],[11],[14],[15],[16],[17],[18],[19],[21],[23],[24],[25],[26],[29],[30],[31],[32],[33],[35]) are the so-called localand Baer-local formations and their generalizations (ω-local and ω-compo-sition formations).Recall that the formation F is said to be: ω-local or ω-saturated if G ∈ Fwhenever G/Op(Φ(G)) ∈ F for any prime p ∈ ω; ω-composition or sol-ubly ω-saturated if G ∈ F whenever G/Φ(Op(G)) ∈ F for any prime p ∈ ω. Alocal formation is an ω-local formation, where ω = P is the set of all primes.Analogously, a composition or Baer-local formation is an ω-composition forma-tion, where ω = P.It is well known that the set of all ω-local formations Lω and the setof all ω-composition formations Cω partially ordered by set inclusion arecomplete lattices (Skiba and Shemetkov in [22] and [27]). Moreover, it wasproved [28] that the lattice of all local formations L is a complete sublatticeof the lattice of all Baer-local formations C.We prove the following generalization of this result in this paper.Theorem The lattice Lω is a complete sublattice of the lattice Cω.All unexplained notations and terminology are standard. The reader isreferred to [1],[5],[21],[20],[22],[26],[27], if necessary.2 PreliminariesRecall that π(G) denotes the set of all prime divisors of the order of a group GFor any collection of groups X we denote by Com(X) the class of all simpleabelian groups A such that A ' H/K, where H/K is a composition factorof G ∈ X.Recall that the subgroup Cp(G) is the intersection of the centralizers ofall the abelian p-chief factors of G (Cp(G) = G if G has no abelian p-chieffactors).The symbols G, Np, Gp ′ , Gω and Sω denote the class of all groups, theclass of all p-groups, the class of all p ′-groups, the class of all ω-groups andthe class of all soluble ω-groups, respectively. For every group class F ⊇ (1),by GF we denote the product of all normal F-subgroups of a group G. Inparticular, we writeOp(G) = GNp , Oω(G) = GGω , Rω(G) = GSω , Op ′,p(G) = GGp ′Np .On sublattices of formations 91Let f be a function of the formf : ω∪ {ω ′}→{formations of groups}. (1)According to [22] and [27] we consider, respectively, two classes of groupsLFω(f) =(G | G/Oω(G) ∈ f(ω ′) and G/Op ′,p(G) ∈ f(p)for all p ∈ ω∩ π(G))andCFω(f) =(G | G/Rω(G) ∈ f(ω ′) and G/Cp(G) ∈ f(p)for all p ∈ ω∩ π(Com(G))).If F is a formation such that F = LFω(f) for a function f of the form (1),then F is said to be ω-local and f is said to be an ω-local satellite of F (see [22]).If F = LFω(f) and f(a) ⊆ F for alla ∈ ω∪ {ω ′},then f is called an inner ω-local satellite of F. The symbol NpF(p) denotes theset of all groups A such that AF(p) is a p-group. According to Remark 1of p.118 in [22], for any ω-local formation F, there exists a unique formationfunction F of the form (1) such that F = LFω(F) and F(p) = NpF(p) ⊆ F forall p ∈ ω. The formation function F is called the canonical ω-local satelliteof F.If F is a formation such that F = CFω(f) for a function f of the form (1),then F is said to beω-composition and f is said to be an ω-composition satelliteof F (see [27]). If F = CFω(f) and f(a) ⊆ F for all a ∈ ω ∪ {ω ′}, then f iscalled an inner ω-composition satellite of F. According to Remark 1 of p.902in [27], for any ω-composition formation F, there exists a unique formationfunction F of the form (1) such that F = CFω(F) and F(p) = NpF(p) ⊆ Ffor all p ∈ ω. The formation function F is called the canonical ω-compositionsatellite of F.Let Θ be a complete lattice of formations. A formation function f ofthe form (1) is called Θ-valued if all its values belong to the lattice Θ. Wedenote by Θωl the set of all formations having an ω-local Θ-valued satel-lite (see [22]); analogously, we denote by Θωc the set of all formations havingan ω-composition Θ-valued satellite (see [27]).By Lω we denote the set of all ω-local formations; by Cω we denotethe set of all ω-composition formations. With respect to inclusion ⊆, thesets Lω and Cω are complete lattices (see p.119 of [22] and p.904 of [27]). Inthe lattice Lω (Cω, respectively) for arbitrary non-empty setΣ = {Hi | i ∈ Λ}92 Nikolay N. Vorob’evof its elements, ⋂i∈ΛHiis the greatest lower bound for Σ in Lω (in Cω, respectively);Lωform(⋃i∈ΛHi)is the least upper bound for Σ in Lω (in CωCωform(⋃i∈ΛHi)is the least upper bound, respectively). Here the symbol Lωform(X)(Cωform(X))denotes the intersection of all ω-local (ω-composition) for-mations containing a collection of groups X.We will use the following results in the proof of the theorem.Lemma 2.1 (see Lemma 4 of [22]) If F = LFω(f) and G/Op(G) ∈ f(p)∩ F forsome p ∈ ω, then G ∈ F.Lemma 2.2 (see Chapter IV, Proposition 1.5 of [5]) Let R/S be a normal sectionof a group G in a formation F, and let K be a normal subgroup of G containedin CG(R/S). With respect to the following action of G/K on R/S :(rS)gK = g−1rgS, r ∈ R, g ∈ Gform the semidirect productH = (R/S)o (G/K).Then H ∈ F.Let Θ be a complete lattice of formations and let X ⊆ F ∈ Θ be a collectionof groups. We write ΘformX to denote the intersection of all formationsof Θ containing all groups of X. For any collection of formations {Fi | i ∈ I}of Θ we write∨Θ(Fi | i ∈ I) = Θform(⋃i∈IFi).Let {fi | i ∈ I} be a collection of Θ-valued functions of the form (1). Then wedenote by ∨Θ(fi | i ∈ I) a function f such thatf(a) = Θform(⋃i∈Ifi(a))On sublattices of formations 93for all a ∈ ω∪ {ω ′}.A complete lattice of formations Θωl is called inductive (see pp.151–152of [26]), if for any collection {Fi | i ∈ I} of formations Fi of Θωl and forany collection {fi | i ∈ I} of inner Θ-valued ω-local satellites fi, where fi isan ω-local satellite of Fi, we have∨Θωl (Fi | i ∈ I) = LFω(∨Θ(fi | i ∈ I)).Lemma 2.3 (see Lemma 4 of [9]) The lattice Lω is inductive.Analogously, a complete lattice of formations Θωc is called inductive [26],if for any collection{Fi | i ∈ I}of formations Fi of Θωc and for any collection{fi | i ∈ I}of inner Θ-valued ω-composition satellites fi, where fi is an ω-compositionsatellite of Fi, we have∨Θωc (Fi | i ∈ I) = CFω(∨Θ(fi | i ∈ I)).Lemma 2.4 (see Theorem of [30] and Theorem 2.1 of [34]) The lattice Cω isinductive.3 Proof of the TheoremLet {Fi | i ∈ I} be a collection of ω-local formations and let Fi be the canoni-cal ω-local satellite of Fi. LetF = ∨Lω(Fi | i ∈ I) and H = ∨Cω(Fi | i ∈ I).It is clear that ⋂i∈IFiis an ω-local formation and this formation is the greatest lower boundfor {Fi | i ∈ I} in Lω. On the other hand, clearly, F is the least upper boundfor {Fi | i ∈ I} in Lω and H is the least upper bound for {Fi | i ∈ I} in Cω.Therefore, in fact, we need only prove that F = H. The inclusion H ⊆ F isevident. Hence, we need only show that F ⊆ H.94 Nikolay N. Vorob’evLet Hi = CFω(Hi), where Hi is an ω-composition satellite such thatHi(a) ={Fi, if a = ω ′,Fi(a), if a = p ∈ ω.We show that Fi = Hi for all i ∈ I.Suppose Hi 6⊆ Fi. Let G be a group of minimal order in Hi \ Fi. Then G isa monolithic group and R = GFi is the monolith of G.If Rω(G) = 1, thenG ' G/1 = G/Rω(G) ∈ Hi(ω ′) = Fi,a contradiction.Hence, Rω(G) 6= 1, i.e., R is a p-group for some p ∈ ω. Since Fi is ω-saturated, it follows that R 6⊆ Φ(G). Consequently, there exists a maximalsubgroup M of G such that R 6 M. Hence, MG = 1, i.e., G is a primitivegroup. Therefore, R = CG(R) = Op ′,p(G). Hence, R = Op(G) = Cp(G).Consequently,G/Op ′,p(G) = G/Cp(G) = G/Op(G) ∈ Hi(p) = Fi(p).Hence, by Lemma 2.1, we have G ∈ Fi, a contradiction. Therefore, Hi ⊆ Fi.Now we show that Fi ⊆ Hi. Assume it is false. Let G be a group ofminimal order in Fi \Hi. Then G is a monolithic group and R = GHi is themonolith of G.Let Oω(G) = 1. Then Rω(G) = 1 andω∩π(Com(G)) = ω∩π(Com(G/R)).Letp ∈ ω∩ π(Com(G)).If R is non-abelian, thenG ' G/1 = G/Rω(G) ∈ Hi(ω ′) = Fi.We have G/R ∈ Hi, by the choice of G. Moreover, Cp(G/R) = Cp(G)/R.Consequently,(G/R)/Cp(G/R) = (G/R)/(Cp(G)/R) ' G/Cp(G) ∈ Hi(p).Thus, G/Cp(G) ∈ Hi(p) for all p ∈ ω∩ π(Com(G)). Hence, G ∈ Hi, a contra-diction.Let R be an abelian ω ′-group. Note thatG ' G/1 = G/Rω(G) ∈ Hi(ω ′) = Fisince Rω(G) = 1. Since G/R ∈ Hi and Cp(G/R) = Cp(G)/R, we haveG/Cp(G) ∈ Hi(p)On sublattices of formations 95for all p ∈ ω∩ π(Com(G)). Hence, G ∈ Hi, a contradiction.Consequently, Oω(G) 6= 1. Let p ∈ π(R) ⊆ ω. If R is non-abelian,then π(Com(R)) = ∅. Hence, Rω(G) = 1. Consequently,G ' G/1 = G/Rω(G) ∈ Hi(ω ′) = Fi.Since G/R ∈ Hi and Cp(G/R) = Cp(G)/R, it follows that G/Cp(G) ∈ Hi(p)for all p ∈ ω∩ π(Com(G)). Hence, G ∈ Hi, a contradiction.Consequently, R is an abelian p-group. We have G/R ∈ Hi, by the choiceof G. Moreover, since R 6 Rω(G), we have Rω(G/R) = Rω(G)/R. It followsthatG/Rω(G) ∈ Hi(ω ′) = Fi.Let T = Ro (G/CG(R)). Since G ∈ Fi, using Lemma 2.2, we have T ∈ Fi.If |T | < |G|, then T ∈ Hi, by the choice of the group G. Hence,G/CG(R) ' T/R = T/CT (R) = T/Cp(T) ∈ Hi(p).Let C∗/R = Cp(G/R). Since G/R ∈ Hi, we have(G/R)/Cp(G/R) = (G/R)/(C∗/R) ' G/C∗ ∈ Hi(p).Moreover, from the above proved we know G/CG(R) ∈ Hi(p). Consequently,G/(C∗ ∩CG(R)) = G/Cp(G) ∈ Hi(p).If q 6= p, then, evidently,Cq(G/R) = Cq(G)/R.Since G/R ∈ Hi, it follows thatG/Cq(G) ∈ Hi(q)for all q ∈ (ω∩ π(Com(G))) \ {p}. Thus,G/Cr(G) ∈ Hi(r)for all r ∈ ω∩ π(Com(G)). Hence, G ∈ Hi, a contradiction.Therefore, |T | = |G|. Hence, R = CG(R). It follows thatR = CG(R) = Cp(G) = Op ′,p(G).Therefore,G/Cp(G) = G/Op ′,p(G) ∈ Fi(p) = Hi(p).96 Nikolay N. Vorob’evMoreover, from the above proved we knowG/Cq(G) ∈ Hi(q)for all q ∈ (ω∩ π(Com(G))) \ {p}. Thus,G/Cr(G) ∈ Hi(r)for all r ∈ ω ∩ π(Com(G)). Hence, G ∈ Hi, a contradiction. Consequent-ly, Fi ⊆ Hi. Thus, Fi = Hi for all i ∈ I.Since by Lemma 2.3 the lattice Lω is inductive, we haveF = ∨Lω(Fi | i ∈ I) = LFω(∨(Fi | i ∈ I)).Since by Lemma 2.4 the lattice Cω is inductive, we haveH = ∨Cω(Fi | i ∈ I) = CFω(∨(Hi | i ∈ I)).Now we are ready to prove the following equality F = H. Clearly, H ⊆ F.Suppose that F 6⊆ H. Let G be a group of minimal order in F \H. Then G is amonolithic group and R = GH is the monolith of G.If Oω(G) = 1, thenG ' G/1 = G/Oω(G) ∈(∨(Fi | i ∈ I))(ω ′) == ∨(Fi(ω′) | i ∈ I) = ∨(Fi | i ∈ I) ⊆ ∨Cω(Fi | i ∈ I) = H,a contradiction.Hence, Oω(G) 6= 1. Let p ∈ π(R) ⊆ ω. If R is non-abelian, thenOp ′,p(G) = 1.Hence, since the canonical ω-local satellite Fi is inner,G ' G/1 = G/Op ′,p(G) ∈(∨(Fi | i ∈ I))(p) == ∨(Fi(p) | i ∈ I) ⊆ ∨(Fi | i ∈ I) ⊆ ∨Cω(Fi | i ∈ I) = H.This contradicts the choice of the group G.Hence, R is an abelian p-group. We have G/R ∈ H, by the choice of G.Moreover, since R 6 Rω(G), we have Rω(G/R) = Rω(G)/R. It follows thatG/Rω(G) ∈(∨(Hi | i ∈ I))(ω ′).Let T = Ro (G/CG(R)). Since G ∈ F, using Lemma 2.2, we have T ∈ F.On sublattices of formations 97If |T | < |G|, then T ∈ H, by the choice of G. Consequently,G/CG(R) ' T/R = T/CT (R) = T/Cp(T) ∈(∨(Hi | i ∈ I))(p).Let C∗/R = Cp(G/R). Since G/R ∈ H, we haveG/C∗ ∈(∨(Hi | i ∈ I))(p).Moreover, from the above proved we knowG/CG(R) ∈(∨(Hi | i ∈ I))(p).Hence,G/Cp(G) ∈(∨(Hi | i ∈ I))(p).If q 6= p, then, evidently, Cq(G/R) = Cq(G)/R. Since G/R ∈ H, it follows thatG/Cq(G) ∈(∨(Hi | i ∈ I))(q)for all q ∈(ω∩ π(Com(G)))\ {p}. Thus,G/Cr(G) ∈(∨(Hi | i ∈ I))(r)for all r ∈ ω∩ π(Com(G)). Hence, G ∈ H, a contradiction.Therefore, |T | = |G|. Consequently, R = CG(R). It follows thatR = CG(R) = Op(G) = Cp(G) = Op ′,p(G).Therefore, since Fi = Hi for all i ∈ I, we haveG/Cp(G) = G/Op ′,p(G) ∈(∨(Fi | i ∈ I))(p) = ∨(Fi(p) | i ∈ I) == ∨(Hi(p) | i ∈ I) =(∨(Hi | i ∈ I))(p).Moreover, from the above proved we knowG/Cq(G) ∈(∨(Hi | i ∈ I))(q)for all q ∈(ω∩ π(Com(G)))\ {p}. Thus,G/Cr(G) ∈(∨(Hi | i ∈ I))(r)for all r ∈ ω∩ π(Com(G)). Hence, G ∈ H. Consequently, F ⊆ H. Thus, F = H,and the theorem is proved.98 Nikolay N. Vorob’evREFERENCES[1] A. Ballester-Bolinches – L.M. Ezquerro: “Classes of FiniteGroups”, Springer, Berlin (2006).[2] A. Ballester-Bolinches – J.É. Pin – X. Soler-Escrivà: “Forma-tions of finite monoids and formal languages: Eilenberg’s variety theo-rem revisited”, Forum Math. 26 (2014), 1737–1761.[3] A. Ballester-Bolinches – J.É. Pin – X. Soler-Escrivà: “Lan-guages associated with saturated formations of groups”, Forum Math.27 (2015), 1471–1505.[4] Ch. Behle – A. 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Vorob’ev: “Algebra of Classes of Finite Groups”, Vitebsk Univer-sity Press, Vitebsk (2012).Nikolay N. Vorob’evDepartment of MathematicsMasherov Vitebsk State University210038, Vitebsk (Belarus)e-mail: vornic2001@mail.ru

On sublattices of the lattice of all ω-composition formations of finite groups

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On sublattices of the lattice of all ω-composition formations of finite groups

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