Some classes of finite groups and mutually permutable products

This paper has been published in Journal of Algebra, 319(8):3343–3351 (2008). Copyright 2008 by Elsevier. The final publication is available at www.sciencedirect.com. http://dx.doi.org/10.1016/j.jalgebra.2007.12.001 http://www.sciencedirect.com/science/article/pii/S002186930700693X Some classes of finite groups and mutually permutable products M. Asaad∗A. Ballester-Bolinches†J.C. Beidleman‡ R. Esteban-Romero§ Abstract This paper is devoted to the study of mutually permutable products of finite groups. A factorised group G=AB is said to be a mutually permutable product of its factors Aand Bwhen each factor permutes with every subgroup of the other factor. We prove that mutually permutable products of Y-groups (groups satisfying a converse of Lagrange’s theorem) and SC-groups (groups whose chief factors are simple) are SC-groups, by means of a local version. Next we show that the product of pairwise mutually permutable Y-groups is supersoluble. Finally, we give a local version of the result stating that when a mutually permutable product of two groups is a PST-group (that is, a group in which every subnormal subgroup permutes with all Sylow subgroups), then both factors are PST-groups. Mathematics Subject Classification (2000): 20D10, 20D40, 20D30 Keywords: mutually permutable product, permutability, Y-group, PST-group, SC-group 1 Introduction and statement of results In this paper we will deal only with finite groups. ∗Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt, email: [email protected] †Departament d’Àlgebra, Universitat de València, Dr. Moliner, 50, E-46100 Burjassot, València, Spain, email: [email protected] ‡Department of Mathematics, University of Kentucky, Lexington, Kentucky 405060027, USA, email: [email protected] §Departament de Matemàtica Aplicada-IMPA, Universitat Politècnica de València, Camí de Vera, s/n, E-46022 València, Spain, email: [email protected] 1 Many group theorists have been worried about what can be said of a group G=G1G2· · · Gmwhich is a product of some pairwise permutable subgroups if some properties of the factors are known. For instance, a wellknown theorem of Kegel and Wielandt [25, 32] says that a product of two nilpotent groups is soluble. The fact that a product of two supersoluble groups is not necessarily supersoluble, even if both factors are normal in the group, motivates the restriction of this question to factorised groups in which both factors are connected by certain stronger permutability properties. The first author and Shaalan introduced in [5] the notion of mutually permutable product G=AB of two subgroups Aand B: in a mutually permutable product, each factor permutes with every subgroup of the other factor. In particular, this situation holds when both factors are normal in the group. Some results about normal products of supersoluble groups were extended to mutually permutable products in [5], for instance, a mutually permutable product G=AB of two supersoluble groups Aand Bis supersoluble whenever G0is nilpotent or one of the factors is nilpotent. They also showed that totally permutable products (that is, every subgroup of each factor permutes with every subgroup of the other factor) of supersoluble groups are supersoluble. Of course, central products and direct products are instances of totally permutable products. Mutually and totally permutable products have been considered as well in [2, 11, 15, 16, 19, 20]. On the other hand, Kegel [26] proved that all subgroups of a group G which permute with all the Sylow subgroups of Gare subnormal. We call these subgroups S-permutable. This motivates the definition of the class of PST-groups or groups in which every subnormal subgroup is S-permutable. Agrawal [1] obtained a characterisation of soluble PST-groups as the groups Gin which the nilpotent residual Lis an abelian normal Hall subgroup of Gand all elements of Ginduce power automorphisms in L. Some interesting subclasses of the class of all PST-groups are the class of all PT-groups (groups in which permutability is a transitive relation, or in which every subnormal subgroup is permutable) and the class of all T-groups (groups in which normality is a transitive relation). These classes of groups have been studied by several authors (for instance, [3, 4, 7, 8, 10, 12, 13, 14, 17, 18, 19, 20, 21, 23, 29, 33]). As a consequence of the theorem of Agrawal [1], soluble PST-groups are supersoluble. Robinson [29] showed that, in the general finite universe, PSTgroups have all their chief factors simple, or, as he says, they are SC-groups. The classification of finite simple groups and the truth of the Schreier conjecture yields the following description of SC-groups: Theorem 1 ([29, Proposition 2.4]).A group Gis an SC-group if and only if 2 there is a perfect normal subgroup Dsuch that G/D is supersoluble, D/ Z(D) is a direct product of G-invariant simple groups, and Z(D)is supersolubly embedded in G(i.e., there is a G-admissible series of Z(D)with cyclic factors). The relation between totally and mutually permutable products and SCgroups has been investigated in [9, 11, 15, 19, 20]. For instance: Theorem 2 ([11, Theorems 2 and 3]).Assume that Gis the mutually permutable product of its subgroups Aand B. Then: 1. If Gis an SC-group, then Aand Bare SC-groups. 2. If Aand Bare SC-groups, then G/ CoreG(A∩B)is an SC-group. Now let us pay attention to the class Yof all groups Gin which for every subgroup Hand all primes qdividing the index |G:H|there exists a subgroup Kof Gsuch that His contained in Kand |K:H|=q. The class Yhas been studied in Chapter 1 and Section 6.1 of [31] and becomes a generalisation of the class of groups satisfying Lagrange’s theorem. These groups can be characterised as follows: Theorem 3. A group Gis a Y-group if, and only if, the nilpotent residual Lof Gis a Hall subgroup of Gand for all subgroups Hof L,G=LNG(H). In [6], it is proved that the class of soluble PST-groups coincides with the class of Y-groups with abelian nilpotent residual. The theory of finite groups has benefited from the local techniques. Given a group theoretical property A, we are interested in finding another weaker property Ap, depending on a prime p, such that a group satisfies Aif and only if it satisfies Apfor all primes p. For instance, p-solubility (pa prime) becomes a good “localisation” of solubility. Local techniques turn out very useful in the study of PST-groups and other related classes. For example, in [21] and [28] the authors have presented some interesting local characterisations of soluble T-groups. A local characterisation of soluble PT-groups appears in [17]. In [3, 12, 13], local characterisations of soluble PST-groups are studied. Let us recall some of these properties: Definition 4. Let pbe a prime number. 1. A p-soluble group Gsatisfies PSTpwhen every p0-perfect subnormal subgroup of Gpermutes with every Hall p0-subgroup of G(see [3]). 2. A group Gsatisfies U∗ pwhen Gis p-supersoluble and all p-chief factors of Gare G-isomorphic when regarded as G-modules (see [3]). 3 3. A group Gsatisfies Ypwhen for every pair of p-subgroups Hand K such that H≤K,His S-permutable in NG(K)([13]). It is shown in [3] and [13] that for p-soluble groups, all three properties are equivalent and so soluble PST-groups are exactly the groups satisfying PSTpfor all primes p. Other local properties for PST-groups in the general finite universe appear in [8]: Definition 5. Let pdenote a prime number. A group Gis said to satisfy Npwhen every non-abelian chief factor of Gof order divisible by pis simple and for each normal subgroup Nof G,p0-elements of Ginduce power automorphisms in Op(G/N). The paper [8] characterises PST-groups as the groups satisfying Npfor all primes p. If we fix a prime p, it is rather clear that a p-soluble group G satisfying property Nphas all p-chief factors G-isomorphic when regarded as G-modules by conjugation. Hence Gis a PSTp-group. Conversely, Assume that Gis a p-soluble PSTp-group. Consider a normal subgroup Nof Gand take a subgroup L/N of Op(G/N). By [3, Lemma 2], G/N is a PSTp-group as well. Then L/N is a subnormal p0-perfect subgroup of G/N, and so L/N permutes with all Hall p0-subgroups of G/N. Let Hbe a Hall p0-subgroup of G. Then HN/N is a Hall p0-subgroup of G/N and L/N is a subnormal Sylow p-subgroup of (L/N)(HN/N). In particular, L/N is normalised by HN/N. This implies that all elements of Hnormalise L. It follows that G is an Np-group. Therefore we have: Lemma 6. Let pbe a prime number. If a group Gis p-soluble, then G satisfies Npif and only if Gsatisfies PSTp. The local method has also been successfully applied to the study of Ygroups in [6] with the definition of the property Zp(pa prime): Definition 7. We say that Gsatisfies Zpwhen for every p-subgroup Xof Gand for every power qmof a prime qdividing |G:XOp0(G)|, there exists a subgroup Kof Gcontaining XOp0(G)such that |K:XOp0(G)|=qm. In [6, Theorem 13], it is proved that property Zpis equivalent to the following one: Theorem 8. Let Gbe a group and let pbe a prime. Then Gsatisfies Zpif and only if Gsatisfies either of the following conditions: 1. Gis p-nilpotent, or 4 2. G(p)/Op0G(p)is a Sylow p-subgroup of G/ Op0G(p)and for every p-subgroup Hof G(p), we have that G= NG(H)G(p). Here X(p)denotes the p-nilpotent residual of a group X, that is, the smallest normal subgroup Nof Xsuch that X/N is p-nilpotent. Theorem 9 ([6, Theorem 15]).A soluble group satisfies Yif and only if it satisfies Zpfor all primes p. In this paper we prove some results on mutually permutable products whose factors belong to some of the above classes. We start with a localisation of SC-groups. Definition 10. Let pbe a prime number. A group Gis said to be an SCp-group whenever every chief factor of Gwhose order is divisible by pis simple. It is clear that Gis an SC-group (i.e., all its chief factors are simple) if and only if Gis and SCp-group for all primes p. In what follows, pwill denote a fixed prime number. The proofs of Theorem 2 can be adapted to prove: Lemma 11. Assume that Gis a mutually permutable product of its subgroups Aand B. 1. If Gis an SCp-group, then Aand Bare SCp-groups. 2. If Aand Bare SCp-groups, then G/ CoreG(A∩B)is an SCp-group. Mutually permutable products of SCp-groups and p-soluble Zp-groups are the object of the next result: Theorem 12. Let G=AB be a mutually permutable product of its subgroups Aand B. Assume that Ais an SCp-group and that Bis a p-soluble Zp-group. Then Gis an SCp-group. The following corollaries follow immediately from Theorem 12: Corollary 13. If Gis a mutually permutable product of an SC-group A and a Y-group B, then Gis an SC-group. In particular, if Gis a mutually permutable product of a supersoluble group Aand a Y-group B, then Gis supersoluble. Let Xbe a class of groups. A class of groups Fis called the Fitting core of Xprovided that whenever if A∈Xand B∈F, and Aand Bare normal subgroups of a group G, then AB ∈X(see [18]). From Corollary 2 of [18] 5 it follows that the class of soluble PST-groups belongs to the Fitting core of the formation of supersoluble groups. In fact from Corollary 13 we obtain a more general statement, mainly: the class Yis contained in the Fitting core of both the formation of supersoluble groups and hence the formation of SC-groups. Corollary 14. If Gis a mutually permutable product of two p-soluble Zpgroups, then Gis p-supersoluble. Corollary 15. If Gis a mutually permutable product of two Y-groups, then Gis supersoluble. Corollary 15 admits the following generalisation: Theorem 16. Let G=G1G2· · · Grbe a group such that G1,G2, . . . ,Grare pairwise mutually permutable subgroups of G. If all Giare Y-groups, then G is supersoluble. We do not know whether a local version of Theorem 16 is true, namely, if all Giare Zp-groups, then Gis p-supersoluble. In [20, Theorem 5], the following result is proved: Theorem 17. Let G=AB be a mutually permutable product of the subgroups Aand B. If Gis a PST-group, then Ais a PST-group. We present in this paper a local version of Theorem 17, from which it follows immediately: Theorem 18. Let Gbe a mutually permutable product of its subgroups A and B. If Gis a SC-group and satisfies Np, then Asatisfies Np. 2 Proofs Proof of Theorem 12. Assume that Gis a counterexample of least order to the result. Since the class of SCp-groups is a formation, then Ghas a unique minimal normal subgroup N. By Lemma 11, Nis a non-cyclic p-subgroup contained in A∩B. The minimal choice of Gimplies that, G/N is an SCpgroup. Set C= CG(N). Then G/C is a mutually permutable product of its subgroups AC/C and BC/C. Since Ais an SCp-group, Nhas an A-composition series with cyclic factors. By [22, IV, 6.9], it follows that AC/C is psupersoluble. The same argument shows that BC/C is p-supersoluble. Since G/C is p-soluble by [19, Corollary 2] and an SCp-group by minimality of G, 6 we conclude that G/C is p-supersoluble. Hence its derived subgroup G0C/C is p-nilpotent by a result of Shemetkov [30]; in particular, G0/N is p-nilpotent. Assume that Nis contained in the Frattini subgroup Φ(G)of G. By [24, 6.6.3], G0is p-nilpotent. Since Nis the unique minimal normal of G and is a p-group, it follows that G0is a p-group. Therefore Ghas a normal Sylow p-subgroup G. This implies that Gis p-soluble. But since Aand B are p-soluble SCp-groups, Aand Bare p-supersoluble. Consequently, Gis p-supersoluble by [19, Corollary 5]. This contradicts the choice of G. Hence Nis not contained in Φ(G). Then Gis a primitive group and Nis a self-centralising minimal normal subgroup of G. Moreover G/N is p-supersoluble. This implies that Aand Bare p-soluble. Since Aand Bare SCp-groups, we have that Aand Bare p-supersoluble. By [19, Theorem 4], K= Op0,p(G) = Op(G) = Nsatisfies that G/K is a p0-group. Hence Nis a normal Sylow p-subgroup of G. In particular, Gis p-soluble and so Aand B are p-supersoluble. By hypothesis, Bis a Zp-group. Then Theorem 8 shows that either Bis p-nilpotent, or B(p)/Op0B(p)is a Sylow p-subgroup of B/ Op0B(p)and for every p-subgroup Hof B(p), we have that B=B(p) NB(H), where B(p) denotes the p-nilpotent residual of B. In the first case, Gis p-supersoluble by [19, Corollary 5]. In the second case, Op0B(p)= 1 because CG(N) = N. Hence B(p)is a p-group and so B(p) = N. Let N1be a minimal normal subgroup of Acontained in N;N1is cyclic because Ais p-supersoluble. Since N1is normalised by Nand by a Hall p0-subgroup of B, we have that N1is normalised by B. It follows that N1is normal in G. This contradiction proves the theorem. The next result is needed in the proof of Theorem 16. Lemma 19. Let Gbe a Y-group with an abelian normal Sylow p-subgroup Pfor a prime p. Then every subgroup of Pis normal in G. Proof. Applying Theorem 3, the nilpotent residual Nof Gis a nilpotent Hall subgroup of G. Therefore there exists a nilpotent subgroup Kof Gsuch that G=NK and gcd(|N|,|K|) = 1 by [22, A, 11.3]. Assume that Pis a subgroup of Nand let Hbe a subgroup of P. Then G=NNG(H)by Theorem 3. Since Nis nilpotent and His normal in P, it follows that N normalises Hand so G= NG(H). Suppose that Pis a subgroup of K. Then, if His a subgroup of P, we have that His normalised by K. Since His subnormal in G, it follows that His a subnormal Sylow p-subgroup of HN. This implies that Nnormalises Hand hence His normal in G=NK. 7 Proof of Theorem 16. Assume that the theorem is false and let Gbe a counterexample with |G|+|G1|+|G2|+· · · +|Gr|minimal. We shall show that this supposition leads to a contradiction. Clearly r > 2by Corollary 15. Besides, the hypotheses of the theorem are inherited by every epimorphic image of G. The minimal choice of Gimplies that every proper quotient of Gis supersoluble. Since the class of all supersoluble groups is a saturated formation, we have that Soc(G)is a supplemented minimal normal subgroup of Gsuch that G/ Soc(G)is supersoluble. Let pdenote the largest prime dividing |G|. Then there exists l∈ {1,2, . . . , r}such that Glcontains a non-trivial Sylow p-subgroup, Plsay. Let i∈ {1, . . . , r},i6=l. Then GlGiis supersoluble by Corollary 15. Hence GlGihas a normal Sylow p-subgroup. In particular, Plis subnormal in GlGi. Applying [27, 7.7.1], Plis subnormal in Gand so Op(G)6= 1. This implies that Soc(G) = F(G)=Op(G)is an abelian minimal normal subgroup of G. Since G/ Op(G)is supersoluble, it follows that Ghas a normal Sylow p-subgroup. Consequently P= F(G)is the unique Sylow p-subgroup of G. Fix an index i∈ {1, . . . , r},i6=l. Then P∩Giis a normal Sylow p-subgroup of Giand so Gi= (P∩Gi)Lifor each Hall p0-subgroup Liof Gi. Moreover, the product GlGiis supersoluble and mutually permutable. Consequently GlLiis a supersoluble subgroup of G. This means that P∩Gl is normalised by Li. Since P∩Glis normal in Gl, we have that P∩Glis normalised by a Hall p0-subgroup Kof G. Since Pis abelian and G=PK, we can conclude that P∩Glis normal in Gand P∩Gl=P. Let Klbe the subgroup generated by all Gi’s, i6=l. Assume that p divides |Kl|. Then, since Klis supersoluble, by the minimal choice of G, we have that Klcontains a normal subgroup Lof order p. By Lemma 19, Lis also normalised by Gl. Consequently Lis normal in G=KlGl, contrary to our supposition. Consequently Klis a p0-group. On the other hand, consider an index j∈ {1, . . . , r},j6=land the subgroup Zgenerated by all Gm’s, m6=j. The minimal choice of Gimplies that Zis supersoluble. Besides, P is a subgroup of Z. Let Nbe a minimal normal subgroup of Zcontained in P. Then |N|=pand NGmis a subgroup of Gas N≤Gl. Since NGmis supersoluble and Gmis a p0-group, it follows that Nis normalised by Gm. Consequently Nis normal in Gand Gis supersoluble. This final contradiction proves the theorem. Our proof of Theorem 18 depends on the following: Lemma 20. Let Nbe a normal subgroup of Gsuch that G/N satisfies Np. If either Nis non-abelian and simple or Nis a p0-group, then Gsatisfies Np. 8