A class of generalized supersoluble groups

Publ. Mat. 49 (2005), 213–223 A CLASS OF GENERALIZED SUPERSOLUBLE GROUPS Adolfo Ballester-Bolinches and Tatiana Pedraza Abstract This paper is devoted to the study of groups Gin the universe c¯ L of all radical locally finite groups with min-pfor all primes psuch that every δ-chief factor of Gis either a cyclic group of prime order or a quasicyclic group. We show that within the universe c¯ Lthis class of groups behaves very much as the class of finite supersoluble groups. 1. Introduction A group Gis said to be supersoluble if it has a finite normal series of cyclic factors. A chief factor of a supersoluble group is cyclic of prime order and a maximal subgroup has prime index. In fact these two properties characterize finite supersoluble groups. It is well known that in the universe of all finite groups the class of supersoluble groups forms a subgroup-closed saturated formation which is intermediate between the classes of nilpotent groups and soluble groups. Several authors have investigated supersoluble groups, not necessarily finite, and generalizations (hypercyclic groups and locally supersoluble groups) extending some results of finite supersoluble groups and establishing connections between these classes of generalized supersoluble groups. In this paper, we introduce a class of generalized supersoluble groups in the universe c¯ Lof all radical locally finite groups with min-pfor all primes p: the class U∗. It is intermediate between the classes of supersoluble c¯ L-groups and hypercyclic c¯ L-groups and it plays a similar role in the class c¯ Las supersoluble groups do in the class of all finite groups. 2000 Mathematics Subject Classification. 20F16, 20F19, 20F50. Key words. Locally finite groups, generalized supersoluble groups, major subgroups. This research is supported by Grant BFM-2001-1667-C03-03, MCyT (Spain) and FEDER (European Union). 214 A. Ballester-Bolinches, T. Pedraza 2. Preliminaries The main purpose of this section is to establish the notation, terminology and some results which will be used throughout this paper. Notation that is not specifically cited here is consistent with that used in [7], [8] and [10]. An infinite group can have insufficient maximal subgroups or even none at all. In order to avoid this situation, Tomkinson [11] introduces the notion of major subgroup. We recall this definition. Let Ube a subgroup of a group Gand consider the properly ascending chains U=U0< U1<···< Uα=G from Uto G, then m(U) is the least upper bound of the types αof all such chains. Clearly m(U) = 1 if and only if Uis a maximal subgroup of G. A proper subgroup Mof Gis said to be a major subgroup of G if m(U) = m(M) whenever M≤U < G. This is a nice extension of the concept of maximal subgroup in the sense that every proper subgroup of a group Gis always contained in a major subgroup of G[11, (2.3)]. In particular, the intersection of all major subgroups of a group G, denoted by µ(G), is a proper subgroup of Gwith properties similar to those of the Frattini subgroup of a finite group. In the sequel, we tacitly assume that all groups belong to the class c¯ L of all radical locally finite groups with min-pfor all primes p. Let Gbe a group and let Mbe a major subgroup of G. Denote MG= CoreG(M). Then G/MGis either a finite soluble primitive group, if Mis a maximal subgroup of G, or a semiprimitive group, if Mis not maximal in G(here, a group Gis said to be semiprimitive if it is the split extension, G= [D]M, of a faithful divisibly irreducible ZM-module D by a finite soluble group M). This result, proved in [2], confirms the importance of major subgroups in the study of the structure of groups and motivates some definitions which are in some sense extensions of well known ones in the finite universe. Definition 1. Suppose that Gis a group and let Mbe a major subgroup of G. We define DM/MG=(Soc(G/MG),if Mis a maximal subgroup of G (G/MG)0,if Mis not a maximal subgroup of G. A Class of Generalized Supersoluble Groups 215 In both cases, DM/MG=F(G/MG), (DM/MG)∩(M/MG) = 1 and CG/MG(DM/MG) = DM/MGfor every major subgroup Mof G. Let Gbe a group and consider two normal subgroups Hand Kof G such that Kis contained in H. Then H/K is called a δ-chief factor of G if H/K is either a minimal normal subgroup of G/K or a divisibly irreducible ZG-module, that is, H/K has no proper infinite G-invariant subgroups. Every δ-chief factor is either an elementary abelian finite p-group for some prime por a direct product of finitely many quasicyclic p-groups for some prime p(see [7, (1.2.4)]). Let Gbe a group. We say that Gis a U∗-group if every δ-chief factor of Gis either a cyclic group of prime order or a quasicyclic group. Obviously the class U∗is a class of generalized supersoluble groups in the universe c¯ Lbecause every finite U∗-group is supersoluble and every supersoluble c¯ L-group is finite and so it is a U∗-group. On the other hand, since every group contains minimal normal subgroups it follows that every U∗-group is hypercyclic (and hence locally supersoluble [1]). Moreover the class U∗is intermediate between the classes of supersoluble groups and hypercyclic groups. On one hand, every quasicyclic p-group, pa prime, is a non-supersoluble U∗-group. On the other hand, let Gbe a cyclic group of order 22. Consider a divisibly irreducible ZG-module A, faithful for G, such that Ais a periodic 2-group (Aalways exists and it is unique up to isomorphism by [9, (3.5)]). Then, applying [9, (5.9)], Ahas rank 2. In particular, the Chernikov group X=AG is not in the class U∗. Moreover, if Kis a normal subgroup of Xsuch that X/K 6= 1 then X/K contains a minimal normal subgroup N/K. Since X/K is a 2-group then it is locally nilpotent and then, by [7, (1.2.6)], N/K is a cyclic group of order 2. We conclude that Xis a hypercyclic group. Let Bbe the class of all groups in which every proper subgroup has a proper normal closure. The class Bhas been introduced and studied in [3], [4]. The results of these papers show that this class is intermediate between nilpotent and locally nilpotent groups, and that it is the natural generalization of the class of finite nilpotent groups from the finite universe to the universe c¯ L. We show that the largest normal B-subgroup of a group is the Fitting subgroup and every δ-chief factor of a group G in Bis central in G. Consequently, every δ-chief factor of Ghas rank one and Gis a U∗-group. Therefore, we obtain in the universe c¯ Lsimilar inclusions to the finite universe for these classes of generalized nilpotent 216 A. Ballester-Bolinches, T. Pedraza groups and generalized supersoluble groups: {Abelian groups} ⊂ {Nilpotent groups} ⊂ B⊂ U∗ and all these inclusions are proper. Let pbe a prime. We say that a group Gis a Bp-group if Gis p-nilpotent and the Sylow p-subgroups of Gare nilpotent. The class Bp is a local version of the class B. The results of the paper [5] show that Bpis a subgroup-closed formation which plays the same role in the universe c¯ Las finite p-nilpotent groups do in the finite one. In particular, every group Ghas a unique largest normal Bp-subgroup denoted by δp0p(G), for every prime p, which is the intersection of the centralizers of all δ-chief factors of Gwhich are p-groups and F(G) = Tpδp0p(G). Let Gbe a U∗-group and let pbe a prime. If H/K is a δ-chief factor of Gwhich is a p-group then G/ CG(H/K) is a cyclic group of order dividing p−1 if p6= 2 or a cyclic group of order dividing 2 if p= 2 by [7, (1.5.18), (1.5.19)]. Therefore G/δp0p(G) is in the class A(p−1) if p6= 2 or in the class A(2) if p= 2, where A(n) denotes the class of all abelian groups with exponent dividing n. Moreover G/Op0p(G)∈A(p−1) for every prime pbecause Op0p(G) is the intersection of the centralizers of all p-chief factors of Gby [7, (6.2.4)]. 3. The results A finite group Gis supersoluble if and only if for every prime p, G/Op0p(G) is abelian of exponent dividing p−1. If Gis a c¯ L-group, that condition does not imply that G∈ U∗(consider for instance the example of a 2-group in Section 2 which is not a U∗-group). Our first result provides a necessary and sufficient condition for a group Gto be aU∗-group. Theorem 1. A group Gis a U∗-group if and only if for every prime p, G/δp0p(G)is in the class A(p−1) if p6= 2 or in the class A(2) if p= 2, where A(n)denotes the class of all abelian groups with exponent dividing n. Proof: Let Gbe a group such that for every prime p,G/δp0p(G) is in the class A(p−1) if p6= 2 or in the class A(2) if p= 2. Consider H/K a chief factor of G. In particular H/K is a p-group for some prime p. Since G/ CG(H/K) is in the class A(p−1) if p6= 2 or in the class A(2) if p= 2 it follows from [8, B, (9.8)] that H/K is a cyclic group of order p. Suppose now that H/K is a divisibly irreducible ZG-module. In particular H/K is a p-group for some prime p. Suppose that p6= 2. A Class of Generalized Supersoluble Groups 217 Then L=G/ CG(H/K) is in the class A(p−1). Since H/K is a divisibly irreducible ZL-module which is faithful for Lit follows from [9, (3.1)] that Lis a cyclic group. Since |L|divides p−1, it follows from [9, (3.4)] that H/K has rank 1, that is, H/K is a quasicyclic p-group. Assume now that p= 2. Then L=G/ CG(H/K) is in the class A(2). Applying [9, (3.1)] we have that Lis trivial or a cyclic group of order 2. Consequently H/K has rank 1 by [9, (3.4)]. We conclude that Gis a U∗-group. Our next results analyze the behaviour of U∗as a class of groups and they are motivated by the well known fact that, in the finite universe, the class of all supersoluble groups is a subgroup-closed saturated formation [8, VII, (2.19)]. Recall that a class Fof groups is said to be a formation if it satisfies the following properties: 1. If G∈Fand Nis a normal subgroup of G, then G/N ∈F. 2. If {Ni}i∈Iis a collection of normal subgroups of Gsuch that G/Ni∈Ffor every i∈Iand Ti∈INi= 1, then G∈F. Theorem 2. The class U∗is a subgroup-closed formation. Proof: It is clear that U∗is closed under taking epimorphic images, that is, U∗is Q-closed. Let {Ni}i∈Ibe a collection of normal subgroups of a group Gsuch that G/Ni∈ U∗for every i∈Iand Ti∈INi= 1. Since G/Ni∈ U∗ we have that G/Ni/(δp0p(G/Ni)) is in the class A(p−1) if p6= 2 or in the class A(2) if p= 2, for all i∈I. Let us denote by GA(n) the A(n)-residual of a group G. Suppose first that p6= 2. Then GA(p−1)Ni/Ni= (G/Ni)A(p−1) ≤δp0p(G/Ni) for all i∈I. In particular, GA(p−1)Ni/Niis in the class Bpfor all i∈I. Since Bpis a formation by [5, Theorem 1] we obtain that GA(p−1) ∈Bp. Consequently G/δp0p(G)∈A(p−1). Suppose now that p= 2. Then GA(2)Ni/Ni= (G/Ni)A(2) ≤δ202(G/Ni) for all i∈I. In particular, GA(2)Ni/Niis in the class B2for all i∈I. Therefore GA(2) ∈B2 and hence G/δ202(G)∈A(2). It follows from Theorem 1 that G∈ U∗. Consequently U∗is a formation. We prove now that U∗is subgroup-closed. Let Gbe a U∗-group and let Hbe a subgroup of G. We have to prove that His also a U∗-group. We split the proof into three cases: Case 1: Gis a Chernikov group such that G0(the radicable part of G) is a quasicyclic p-group for some prime p. Since H0is a divisible subgroup of G0, then either H0= 1 or H0=G0. First we assume that H0=G0. 218 A. Ballester-Bolinches, T. Pedraza Let A/B be a δ-chief factor of H. Since the class U∗is Q-closed, we may assume that B= 1. If Ais a divisibly irreducible ZH-module, then Ais a quasicyclic p-group. Suppose now that Ais a minimal normal subgroup of H. Then either A≤H0or A∩H0= 1. If Ais contained in H0, then Ais a cyclic group of prime order pbecause it is an elementary abelian. Assume that A∩H0= 1. Since AH0/H0is a minimal normal subgroup of H/H0and H/H0is a finite supersoluble group, it follows that AH0/H0is a cyclic group of prime order and so is A. Suppose now that H0= 1, that is, His finite. If A/B is a chief factor of G, then A/B is a cyclic group of prime order. In particular (H∩A)/(H∩B) is either trivial or a cyclic group of prime order. Hence a chief series of G intersected with Hwill yield, after deleting redundant terms, a chief series of H(of finite length) with cyclic factors. Therefore H∈ U∗. Case 2: Gis a Chernikov group with Op0(G) = 1 for some prime p. Then G=G0M, where Mis a finite subgroup of G. We can certainly assume that G06= 1, since otherwise Gis a finite supersoluble group and so the result follows. Therefore G0is a divisible abelian p-group of finite rank. By [12, (1.3)] there is a finite normal subgroup Cof Gcontained in G0 such that G0/C is a direct product of divisibly irreducible ZG-modules, say G0/C = (G1/C)×(G2/C)× · · · × (Gn/C). Since Gi/C is a δ-chief factor of Gand G∈ U∗, we have that Gi/C is a quasicyclic p-group for all i∈ {1,...,n}. Let us first assume that C= 1 and denote X1=G2×G3×· · ·×Gnand Xi=G1×· · ·× Gi−1×Gi+1 × · · · × Gnfor all i > 1. It is clear that Xiis normal in Gfor all iand Ti≥1Xi= 1. Moreover G/Xi∼ =GiMand (GiM)0=Giis a quasicyclic p-group. Applying Case 1, we have that H/(H∩Xi) is a U∗-group for every i. Since U∗is a formation, we conclude that H∈ U∗. Assume now that C6= 1. By the above argument, HC/C is a U∗-group. We prove that HC ∈ U∗. Let A/B be a δ-chief factor of HC. There is no loss of generality in assuming that B= 1. Suppose that Ais a minimal normal subgroup of HC. Then either A∩C= 1 or A∩C=A. If A∩C= 1, it follows that AC/C is a minimal normal subgroup of HC/C ∈ U∗and so Ais a cyclic group of prime order. Assume that A≤C. Let 1 = C0CC1CC2C···CCs=C be part of a chief series of Gpassing through C, that is, Ci/Ci−1is a minimal normal subgroup of G/Ci−1for all i∈ {1,...,s}. Since G∈ U∗, we have that Ci/Ci−1is a cyclic group of prime order for all i. On the other hand, since 1 6=A≤Cthere exists j∈ {1,...,s}such that A Class of Generalized Supersoluble Groups 219 A≤Cjand ACj−1. In particular, 1 6=ACj−1/Cj−1≤Cj/Cj−1 and then A∼ =ACj−1/Cj−1is a cyclic group of prime order. Suppose now that Ais a divisibly irreducible Z(HC)-module. Then A∩Cis a proper finite subgroup of A. This implies that A/(A∩C)∼ =AC/C is a divisibly irreducible Z(HC/C)-module and so it is a quasicyclic group. Consequently Ais a quasicyclic group. We have proved that HC is a U∗-group. Since U∗is a formation and Cis an abelian normal subgroup of HC, it follows from [2, Lemma 2] that H∈ U∗, which is our claim. General case: By [7, (2.5.13)], G/Op0(G) is a Chernikov U∗-group for every prime p. Applying Case 2, we have that H/(H∩Op0(G)) is a U∗- group for every prime p. Since Tp(H∩Op0(G)) = 1 and U∗is a formation, we conclude that Hbelongs to U∗, which completes the proof. Bearing in mind that the class of supersoluble groups in the finite universe is saturated, one can wonder if U∗could enjoy this property in our universe. The answer is negative as the following example shows: Example 1. Let Xbe the regular wreath product of a quasicyclic p-group and a cyclic group of order p, where pis a prime number. Consider G=X/ Z(X). Then G=G0Mis a semiprimitive group such that G0is isomorphic to a direct product of p−1 quasicyclic p-groups and Mis a cyclic group of order p. Moreover, the group Gcan be expressed as G=Si≥1Gi, where Gi= Ωi(G0)M, for each natural number iand {Gi:i≥1}is an ascending chain of subgroups of G. Notice that Gi is a finite p-group and hence a supersoluble group for each i≥1. However, if p6= 2, Gis not a U∗-group because G0is a divisibly irreducible ZG-module which is not quasicyclic. Applying [2, Theorem A], we have that U∗is not saturated. Nevertheless, U∗is closed under taking extensions by Tomkinson’s Frattini-like subgroup as our next result shows. Theorem 3. Let Nbe a normal subgroup of a group Gsuch that N/µ(G) is a U∗-group. Then N∈ U∗. Proof: Assume first that Gis a Chernikov group. Then µ(G) is finite by [12, (1.2)]. On the other hand, for every prime p, (N/µ(G))/Op0p(N/µ(G)) is abelian of exponent dividing p−1 because N/µ(G)∈ U∗. Moreover, the arguments used in [11, (5.1)] allow us to show that Op0p(N/µ(G)) = Op0p(N)/µ(G). Thus, for every prime p,N/Op0p(N) is abelian of exponent dividing p−1. Let H/K be a δ-chief factor of N. Then either Hµ(G)/Kµ(G) is N-isomorphic to H/K or Hµ(G) = Kµ(G) and H∩µ(G)/K ∩µ(G) is 220 A. Ballester-Bolinches, T. Pedraza N-isomorphic to H/K. Consequently, there is no loss of generality in assuming that either K≤H≤µ(G) or µ(G)≤K≤H. In the first case, we have that H/K is finite and therefore H/K is a p-chief factor of N for some prime p. In particular, AutN(H/K)∼ =N/ CN(H/K) is finite and abelian of exponent dividing p−1. Applying [8, B, (9.8)], H/K is a cyclic group of order p. Assume now that µ(G)≤K≤H. Then H/K is isomorphic to a δ-chief factor of N/µ(G)∈ U∗and consequently H/K is a cyclic group of prime order or a quasicyclic group. We conclude that N∈ U∗. In the general case, applying [7, (2.5.13)], G/Op0(G) is a Chernikov group for every prime p. Moreover, since N/µ(G)∈ U∗and µ(G)Op0(G)/Op0(G)≤µ(G/Op0(G)) it follows that NOp0(G)/Op0(G) is a normal subgroup of G/Op0(G) satisfying the hypothesis of the theorem. By the above argument, we obtain that N/(N∩Op0(G)) is a U∗-group, for every prime p. Since U∗is a formation, we conclude that Nbelongs to U∗, as required. Corollary 1. Let Gbe a group. Then G/µ(G)∈U∗if and only if G∈U∗. If Mis a major subgroup of a group Gsuch that DM/MGis a cyclic group of prime order or a quasicyclic group then G/MGis a U∗-group. This fact motivates the following: Definition 2. Let Gbe a group and let Mbe a major subgroup of G. We define the extended index of Min G, denoted by qG(M), as the rank of DM/MG. Note that Mis a major subgroup of G, then qG(M)6= 1 if and only if G/MGis not in U∗. Consequently the following result holds. Corollary 2. Let Gbe a group. Then G∈ U∗if and only if the extended index of Mis 1for every major subgroup Mof G. Note that the above result extends a well known result of Huppert [8, VII, (2.2)]. Bathia [6] proved that the intersection of all maximal subgroups of a finite group of composite index is a supersoluble characteristic subgroup of the group. Our aim now is to obtain a similar result in our universe. Definition 3. Let Gbe a group. We define L(G) = \{M:Mis a major subgroup of Gsuch that qG(M)6= 1} =\{M:Mis a major subgroup of Gsuch that G/MG/∈ U∗}. We stipulate that L(G) = Gif the above set of major subgroups is empty. A Class of Generalized Supersoluble Groups 221 Theorem 4. Let Gbe a group. Then L(G)is a characteristic U∗-subgroup of G. Proof: It is clear that L(G) is a characteristic subgroup of G. We prove now that L(G) belongs to U∗. If the above set of major subgroups is empty then L(G) = G∈ U∗by Corollary 1. Then we may assume that this set is non-empty. Assume first that µ(G) = 1. Let L(G)U∗ be the U∗-residual of L(G), that is, the intersection of all normal subgroups Nof L(G) such that L(G)/N ∈ U∗. By Theorem 1, we have that L(G)/L(G)U∗is a U∗-group. We will show that L(G)U∗is a subgroup of µ(G). Obviously L(G)U∗is contained in every major subgroup Mof G such that G/MG/∈ U∗. Suppose now that Mis a major subgroup of G such that G/MG∈ U∗. Then, the U∗-residual of G,GU∗, is contained in MG. Since U∗is subgroup-closed, it follows that L(G)U∗≤GU∗. Consequently, L(G)U∗is a subgroup of M. This implies that L(G)U∗is contained in every major subgroup of Gand hence L(G)U∗≤µ(G) = 1. We conclude that L(G)∈ U∗. Suppose now that µ(G)6= 1 and consider G/µ(G). By the above argument we have that L(G/µ(G)) ∈ U∗. Moreover, L(G/µ(G)) = L(G)/µ(G). Therefore it follows from Theorem 3 that L(G) is a U∗-group. It is well known that if Gis a finite supersoluble group, then Ghas a normal Sylow p0-subgroup [8, VII, (2.1)] for the smallest prime pdividing the order of G. Moreover, the derived subgroup of Gis nilpotent. The corresponding versions in our universe for the class U∗are the following: Theorem 5. Let Gbe a U∗-group. Then: a) Ghas a normal Sylow p0-subgroup for the smallest prime pdividing the orders of the elements of G. b) G0≤F(G); in particular, G0belongs to B. Proof: a) Let Mbe a major subgroup of Gand denote MG= CoreG(M). Suppose that the result is true for G/MGfor every major subgroup Mof G. Let Qbe a Sylow p0-subgroup of G. Since all p0-subgroups of Gare conjugate and QMG/MGis a Sylow p0-subgroup of G/MGby [7], it follows that G= (NG(Q))MGfor every major subgroup Mof G. Since every proper subgroup of Gis contained in a major subgroup of G, it follows that G= NG(Q). That is, Gcontains a normal Sylow p0-subgroup. Hence there is no loss of generality in assuming that MG= 1 and then Gis either a finite