On finite soluble groups in which Sylow permutability is a transitive relation

This paper has been published by Springer-Verlag and Akadémiai Kiadó in Acta Mathematica Hungarica, 101(3):193–202 (2003). The final publication is available at http://www.springerlink.com http://dx.doi.org/10.1023/B:AMHU.0000003903.71033.fc. On finite soluble groups in which Sylow permutability is a transitive relation∗ A. Ballester-Bolinches Departament d’` Algebra Universitat de Val`encia Dr. Moliner, 50 E-46100 Burjassot (Val`encia) Spain email: [email protected] R. Esteban-Romero Departament de Matem`atica Aplicada Universitat Polit`ecnica de Val`encia Cam´ı de Vera, s/n E-46022 Val`encia Spain email: [email protected] 20th January 2013 Abstract A characterisation of finite soluble groups in which Sylow permutability is a transitive relation by means of subgroup embedding properties enjoyed by all the subgroups is proved in the paper. The key point is an extension of a subnormality criterion due to Wielandt. 1 Introduction and statements of results One of the principal objectives of this paper is to give characterisations of finite soluble groups in which Sylow permutability is a transitive relation by means of two subgroup embedding properties, weak S-permutability and Ssubpermutiser condition, which will be defined below. Our approach involves an analysis of the relation between the above properties and Sylow permutability. In this context, a nice extension of a well-known subnormality criterion due to Wielandt turns out to be crucial. Recall that a subgroup Hof a finite group Gis said to be S-permutable in G if Hpermutes with all Sylow subgroups of G. According to a theorem of Kegel [10], every S-permutable subgroup is subnormal. A group Gis said to be a PST-group if every subnormal subgroup of Gis S-permutable in G. Subclasses of PST-groups are the class of PT-groups or groups in which permutability is transitive and the class of T-groups or groups in which normality is transitive. There are several characterisations of finite soluble T-groups, PT-groups and PST-groups in terms of normal structure and Sylow structure ([1, 2, 3, 4, 5, 7, 9, 12]). Theorem 3 of [4] explains clearly the parallelism between these characterisations. Roughly speaking, one can get a T-characterisation (respectively, a ∗Supported by Grant BFM2001-1667-C03-03, MCyT, Spain, and FEDER, European Union Keywords and phrases: finite groups, permutability, subnormality Mathematics Subject Classification (2000): 20D35, 20D10, 20D20 1 PT-characterisation) from a PST-characterisation just by adding ‘Dedekind’ (respectively, ‘modular’) to the Sylow subgroups and substituting ‘S-permutable’ by ‘normal’ (respectively, ‘permutable’). Recently, Bianchi, Gillio Berta Mauri, Herzog and Verardi [6] present a new characterisation of soluble T-groups using the following embedding property: A subgroup Hof Gis said to be an H-subgroup of Gif for all g∈G, NG(H)∩Hg≤H. They prove: Theorem 1 ([6, Theorem 10]).A group Gis a soluble T-group if and only if every subgroup of Gis an H-subgroup. The above embedding property is closely related to the weak normality, studied by the authors in [3]: A subgroup Hof Gis called weakly normal in Gif Hg≤NG(H) implies that g∈NG(H). If His weakly normal in Gand His normal in a subgroup Kof G, then NG(K) is contained in NG(H). This fact is crucial in the proof of [6, Theorem 10] and is a subgroup embedding property also studied in [3]: A subgroup Hof Gis said to satisfy the subnormaliser condition if for every subgroup Kof Gsuch that HEK, it follows that NG(K)≤NG(H). Although neither a weakly normal subgroup is an H-subgroup nor a subgroup satisfying the subnormaliser condition is weakly normal ([3, Example 2]), we have: Theorem 2 ([3]).The following statements are equivalent: 1. Gis a soluble T-group. 2. Every subgroup of Gis weakly normal in G. 3. Every p-subgroup of Gis weakly normal in Gfor all primes p. 4. Every subgroup of Gsatisfies the subnormaliser condition in G. 5. Every p-subgroup of Gsatisfies the subnormaliser condition in Gfor all primes p. In view of the parallelism between the characterisations of finite soluble T-, PTand PST-groups in terms of the normal structure and Sylow structure, it is of interest to investigate the following situation: Is it possible to define PTand PST-versions of the above embedding properties to get the PTand PST-versions of Theorems 1 and 2? This paper tries to give the complete answer to this question. Let us begin with the following elementary equivalences: 2 •A subgroup Hof a group Gis weakly normal in Gif and only if Hsatisfies the following property: if g∈Gand His normal in hH, Hgi, then His normal in hH, gi. •A subgroup Hof a group Gsatisfies the subnormaliser condition in Gif and only if for every subgroup Kof Gsuch that His normal in Kand for every element x∈Gsuch that Kis normal in hK, xi, we have that H is normal in hH, xi. Therefore it seems natural to consider the following embedding properties, which can be regarded as the PST-versions of the abovementioned ones: Definition 1. We say that a subgroup Hof a group Gis weakly S-permutable in Gwhen the following condition holds: If g∈Gand His S-permutable in hH, Hgi, then His S-permutable in hH, gi. Definition 2. We say that a subgroup Hof a group Gsatisfies the S-subpermutiser condition in Gwhen the following condition holds: If His S-permutable in Kand xis an element of Gsuch that Kis S-permutable in hK, xi, then His S-permutable in hH, xi. Note that there exist subgroups Hsuch that His S-permutable in hH, Hgi for all g∈G, but His not S-permutable in G, as Example 1 shows. Example 1.Consider the group G= Σ4, the symmetric group of degree 4, and H=h(1,2)(3,4)i. For every g∈G,hH, Hgi ≤ Soc(G). In fact, if g /∈NG(H), hH, Hgi= Soc(G)EG, hence His S-permutable in hH, Hgi, but His not Spermutable in hH, gifor some g∈G, e g, g= (1,2,3) (notice that hH, gi=A4). In particular, His not S-permutable in G. Clearly S-permutable subgroups are weakly S-permutable. Maximal subgroups, Sylow subgroups and self-normalising subgroups are weakly S-permutable, too. The following proposition shows the relation between the above properties and the corresponding T-versions. Proposition 1. Let Hbe a subgroup of a group G. Then: 1. If His weakly normal in G, then His weakly S-permutable in G. 2. If Hsatisfies the subnormaliser condition in G, then Hsatisfies the Ssubpermutiser condition in G. Obviously the next step will be to analyse the relation between weak S-permutability and S-subpermutiser condition. There exist subgroups satisfying the S-subpermutiser condition which are not weakly S-permutable (see Example 2 below). However, we prove in the following that weak S-permutability implies the S-subpermutiser condition. The strategy used is the following: It is clear that a subgroup Hof a group Gis normal (respectively, permutable) in Gif and only if His normal in hH, gifor every g∈G. Less trivial is the following result of Wielandt: 3 Theorem 3. For a subgroup Hof a group G, the following statements are equivalent: 1. His subnormal in G. 2. His subnormal in hH, Hgifor all g∈G. 3. His subnormal in hH, gifor all g∈G. Example 1 shows that the equivalence between 1 and 2 does not hold neither for normality, nor permutability nor S-permutability. Nevertheless, the equivalence between 1 and 3, already noted above for normality and permutability, also holds for S-permutability, and it is a key result which helps to relate weak Spermutability and S-subpermutiser condition to S-permutability. Theorem A. A subgroup Hof a group Gis S-permutable in Gif and only if His S-permutable in hH, gifor every g∈G. Applying Theorem A we have: Corollary 1. If Hsatisfies the S-subpermutiser condition in a group Gand H is a subnormal subgroup of a subgroup Kof G, then His S-permutable in K. Corollary 2. If His weakly S-permutable in G, then Hsatisfies the S-subpermutiser condition in G. Next we deal with certain localisations of PST-, PTand T-groups. Fix a prime p. Robinson [11] introduced the class Cpof all groups Gsuch that each subgroup of every Sylow p-subgroup Pof Gis normal in NG(P). He proves that a group Gis a soluble T-group if and only if it belongs to the class Cpfor all primes p. The PT-version of the class Cpis the class Xpintroduced by Beidleman, Brewster and Robinson in [5]: a group Gbelongs to Xpif and only if each subgroup of every Sylow p-subgroup Pof Gis permutable in NG(P). A group Gis a soluble PT-group if and only if Gbelongs to the class Xpfor all primes p([5, Theorem A]). The PST-version of the above classes is the class Yp introduced by the authors in [4]: a group Gbelongs to Ypif and only if when Hand Kare p-subgroups of Gsuch that H≤K, then His S-permutable in NG(K). A group Gis a soluble PST-group if and only if Gbelongs to the class Ypfor all primes p([4, Theorem 4]). Bryce and Cossey [7] characterise in the soluble universe the groups in the class Cpas the groups Gin which every p0-perfect subnormal subgroup of Gis normal in G. We also prove in [3] that a soluble group Gbelongs to the class Cpif and only if every p0-perfect subgroup is weakly normal in G. It is natural then to ask for the relation between the class Ypand weak Spermutability and S-subpermutiser condition. First of all, note that there exist groups in the class Ypwith p0-perfect subnormal subgroups which are neither weakly S-permutable nor satisfy the S-subpermutiser condition (see Section 3). The best result we get is: Theorem B. Let Gbe a group. The following statements are equivalent: 1. Gis a Yp-group. 2. Every p-subgroup of Gsatisfies the S-subpermutiser condition in G. 4 With the above results at hand, we are able to prove the following characterisations of soluble PST-groups. Theorem C. Let Gbe a group. The following statements are equivalent: 1. Gis a soluble PST-group. 2. Every subgroup of Gis weakly S-permutable in G. 3. For every prime number p, every p-subgroup of Gis weakly S-permutable in G. 4. Every subgroup of Gsatisfies the S-subpermutiser condition in G. 5. For every prime number p, every p-subgroup of Gsatisfies the S-subpermutiser condition in G. 2 Proofs Proof of Proposition 1. 1. Suppose that His a weakly normal subgroup of G. Let gbe an element of Gsuch that His S-permutable in hH, Hgi. By Kegel’s Theorem [10] we know that His subnormal in hH, Hgi. Now applying [3, Lemma 1] we have that His normal in hH, Hgi. The weak normality of Hin Gimplies that His normal in hH, giand, in particular, His S-permutable in hH, gi. Consequently, His weakly S-permutable in G. With the same arguments to those used in the proof of statement 1 and applying Kegel’s theorem and [3, Lemma 1], we have that each subgroup satisfying the subnormaliser condition in Galso satisfies the S-subpermutiser condition in G. Proof of Theorem A. Suppose that Gis a group of minimal order with a subgroup Hsuch that His S-permutable in hH, gifor every g∈G, but His not S-permutable in G. Since His a subnormal subgroup of hH, gifor every g∈G, from Theorem 3 it follows that His a subnormal subgroup of G. Let Mbe a maximal normal subgroup of Gcontaining H. Since His not S-permutable in G, there exists a prime pand a Sylow p-subgroup Pof Gsuch that Pdoes not permute with H. Suppose that there exists a maximal subgroup M1of Gsuch that H≤M1 and Mis not contained in M1. Then MM1=G. From the minimality of G, it follows that His S-permutable in Mand M1. Moreover, there exists a Sylow p-subgroup Qof Mand a Sylow p-subgroup Q1of M1such that their product QQ1=P0is a Sylow p-subgroup of G. Then Hpermutes with both Qand Q1, hence Hpermutes with P0. Consider a minimal normal subgroup Nof G contained in M. By minimality of G,HN/N permutes with PN/N, hence HN permutes with Pand P(HN) is a subgroup of G. If P(HN) is a proper subgroup of G, then Hpermutes with P, a contradiction. Consequently we have that P(HN) = G. There exists an element x∈Gsuch that P0=Px, and xcan be expressed as x=x1x2, with x1∈Pand x2∈HN. Therefore P0=Px=Px2. Hence Hpermutes with Px2, or, equivalently, Hx−1 2permutes with P. Since H is a subnormal subgroup of G, by a theorem of Wielandt [8, A,14.3] we have that Soc(G) normalises each subnormal subgroup of G. In particular, His a 5 normal subgroup of HN, and since x2∈HN, we have that H=Hx−1 2. This implies that Hpermutes with P, a contradiction. Consequently, if M1is a maximal subgroup of Gcontaining H, then M≤M1. Since P(HN) = Gand HN ≤M, it follows that |G:M|is a power of p. Hence all maximal subgroups of G/M are normal. Thus Mis actually a maximal subgroup, and it is the unique maximal subgroup of Gcontaining H. Therefore if x∈G\M, we have that hH, xi=G: otherwise there would exist another maximal subgroup of G containing H. From the hypothesis, His S-permutable in hH, xi=G, the final contradiction. The converse is clear. Note by Theorem A that a subgroup Hof a group Gsatisfies the S-subpermutiser condition in Gif and only if Hsatisfies the following property: If His S-permutable in Kand Kis S-permutable in L, then His S-permutable in L. Proof of Corollary 1. Suppose that Hsatisfies the S-subpermutiser condition in Gand that His subnormal in a subgroup Kof G. Arguing by induction we can suppose, without loss of generality, that His S-permutable in a proper normal subgroup Lof K. Consider g∈K. Since His S-permutable in Land Lis Spermutable in hL, gi, from the S-subpermutiser condition we have that His Spermutable in hH, gi. Since this happens for every g∈K, from Theorem A we obtain that His an S-permutable subgroup of K. Proof of Corollary 2. Assume that His a weakly S-permutable subgroup of G. Let Kbe a subgroup of Gsuch that His S-permutable in K. Suppose in addition that xis an element of Gsuch that Kis S-permutable in hK, xi. By Kegel’s theorem, we have that His subnormal in hK, xi. By Corollary 1 we obtain that His S-permutable in hK, xi, as desired. Proof of Theorem B. Suppose that every p-subgroup of Gsatisfies the S-subpermutiser condition in G. Suppose that H≤L≤P, where Pis a Sylow p-subgroup of G. Since His a subnormal subgroup of NG(L) and Hsatisfies the S-subpermutiser condition in G, we have that His S-permutable in NG(L) by Corollary 1. Therefore Gis in the class Yp. Now suppose that Gis in the class Yp. Assume that His an S-permutable psubgroup of K, and Kis an S-permutable subgroup of L. Arguing by induction, we can suppose that H≤KELand that His S-permutable in K. Since G belongs to the class Yp,His S-permutable in NG(K), which contains L. In particular, His S-permutable in L. Proof of Theorem C. Let us see that 1 implies 2. Suppose that Gis a soluble PST-group. Applying the results of [1], G=AB, where Ais the nilpotent residual of G,Ais abelian of odd order, |A|and |B|are coprime and every subgroup of Anormal in G. Let g∈Gand H≤Gsuch that His S-permutable in hH, Hgi. We can suppose that Gis not nilpotent, and so A6= 1. Let Nbe a minimal normal subgroup of Gsuch that N≤A. By minimality of G,HN/N is weakly S-permutable in G/N. Hence HN/N is S-permutable in hH, giN/N. Consequently HN is S-permutable in hH, giN. If hH, giis a proper subgroup of G, then His S-permutable in hH, gi. Therefore G=hH, giand HN is Spermutable in G. This implies that HN is a subnormal subgroup of G. 6 Assume that His not weakly S-permutable and let pbe a prime number dividing |G|and Pa Sylow p-subgroup of Gsuch that Hdoes not permute with P. If (HN)Pis a proper subgroup of G, then Hpermutes with Pby induction. Consequently, G= (HN)P. Suppose that pdivides |A|, then P≤Aand Pis a normal subgroup of G. Hence Hpermutes with P, a contradiction. Therefore |P|and |A|are coprime. Moreover, CoreG(H) = 1. Thus H∩A= 1 and |H| and |A|are coprime. As a consequence, if πis the set of primes dividing |A| and nπis the π-part of the number n, then |G|π=|HN|π|P|π |HN ∩P|π =|HN|π=|N|π and hence A=N. Let us denote T=hH, Hgiand let qbe the prime dividing |N|. If |T|q6= 1, then N∩Tis a nontrivial normal subgroup of G. Hence N≤T. Since His S-permutable in T, we have that His a subnormal subgroup of Tand so His subnormal in HN. Therefore His a subnormal subgroup of G. Since Gis a PST-group, we have that His S-permutable in G, a contradiction. Therefore |T|q= 1. We can suppose that T≤B. The element gcan be expressed as g=bn, with b∈Band n∈N=hxi, with o(x) = p(notice that Gis supersoluble). If n= 1, then His S-permutable in hH, bi=hH, gi, because Bis nilpotent. Hence n6= 1 and N=hniand Hg≤Bg=Bn, therefore Hg≤B∩Bn=CB(n) (see [8, A,16.3]). Consequently Hg≤CG(n), whence Hb≤CG(n)n−1 =CG(n). This implies that Hb≤CG(N) and so HbNis a nilpotent group. But in this case Hbis a subnormal subgroup of G, because HbNis a subnormal subgroup of G. Therefore His a subnormal subgroup of G. Since Gis a PST-group, we have that His S-permutable in G, the final contradiction. It is obvious that 2 implies 3 and that 4 implies 5. From Proposition 1, it follows that 2 implies 4 and that 3 implies 5. From Theorem B and [4, Theorem 5], it follows that 5 implies 1. This completes the proof. 3 An example Example 2.Consider P=hx, y |x2=y8= 1, yx=y5i, a modular group of order 16. Phas an irreducible and faithful module over the field of 17 elements, V=hw1, w2i, such that the action of Pis described by wx 1=w2,wx 2=w1, wy 1=w9 1,wy 2=w8 2. We construct the semidirect product G= [V]P. We observe that xcentralises the element w1w2. Let g=w1w2y. Let H=hxi. We have that Hg=hxyi=hxy4i ≤ P. Consequently the subgroup H=hxiis Spermutable in hH, Hgi. But His not S-permutable in hH, gi=G: it suffices to see that Hdoes not permute with, e g, Pw1. It is clear that Gis a 2-nilpotent group, and so Gbelongs to the class Y2 by [4, Theorem 5]. Applying Theorem B, all 2-subgroups of G, in particular H, satisfy the S-subpermutiser condition in G(the reader is invited to prove directly that Hsatisfies the S-subpermutiser condition in G). Consider the subgroup L=hx, w1w−1 2i. Then Lis a 20-perfect subnormal subgroup of Gwhich is not permutable with P. However, Lis S-permutable in M=hx, y2, w1, w2iEGand Mis S-permutable in G=hM, gi, but Lis not S7 permutable in G=hL, gi. It follows that Ldoes not satisfy the S-subpermutiser condition in G. 4 Postscript: An extension to P T -groups In this section we introduce two new embedding properties useful to give characterisations of PT-groups. Definition 3. We say that a subgroup Hof a group Gis weakly permutable when the following condition holds: If His permutable in hH, Hgi, then His permutable in hH, gi. Definition 4. We say that a subgroup Hof a group Gsatisfies the subpermutiser condition in Gwhen the following condition holds: If His permutable in Kand xis an element of Gsuch that Kis permutable in hK, xi, then His permutable in hH, xi. Weak permutability and the subpermutiser condition extend weak normality and the subnormaliser condition, respectively, to permutability. The following results hold: Theorem 4. 1. If His a weakly normal subgroup of G, then His a weakly permutable subgroup of G. 2. If His a weakly permutable subgroup of G, then His a weakly S-permutable subgroup of G. 3. If Hsatisfies the subnormaliser condition in G, then Hsatisfies the subpermutiser condition in G. 4. If Hsatisfies the subpermutiser condition in G, then Hsatisfies the Ssubpermutiser condition in G. 5. If His a weakly permutable subgroup of G, then Hsatisfies the subpermutiser condition in G. 6. If His weakly permutable in Gand His a subnormal subgroup of a subgroup Kof G, then His permutable in K. 7. If Hsatisfies the subpermutiser condition in Gand His a subnormal subgroup of a subgroup Kof G, then His permutable in K. We can give now PT-versions of Theorem B and Theorem C. Theorem D. Let Gbe a group. The following statements are equivalent: 1. Gbelongs to Xp. 2. Every p-subgroup of Gsatisfies the subpermutiser condition. Theorem E. Let Gbe a group. The following statements are equivalent: 1. Gis a soluble PT-group. 8