Sylow permutable subnormal subgroups of finite groups

This paper has been published in Journal of Algebra, 251(2):727–738 (2002). The final publication is available at www.sciencedirect.com. http://dx.doi.org/10.1006/jabr.2001.9138 Sylow permutable subnormal subgroups of finite groups∗ 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]v.es Dedicated to John Cossey on the occasion of his sixtieth birthday Abstract An extension of the well-known Frobenius’ criterion of p-nilpotence in groups with modular Sylow p-subgroups is proved in the paper. This result is useful to get information about the classes of groups in which every subnormal subgroup is permutable and Sylow permutable. ∗Supported by Proyecto PB97-0674 and Proyecto PB97-0604-C02-02 from DGICYT, Ministerio de Educaci´on y Ciencia 1 1 Introduction and statements of results Throughout the paper, the word group means finite group. A celebrated theorem of Frobenius ([9, Satz IV.5.8]) asserts that if pis a prime and Gis a group such that NG(H) is p-nilpotent for every p-subgroup Hof G, then Gis p-nilpotent. Our first main result can be considered as an extension of Frobenius’ theorem in groups with modular Sylow p-subgroups. Theorem 1. Let pbe a prime and let Gbe a group with a modular Sylow p-subgroup P. Then Gis p-nilpotent if and only if NG(P)is p-nilpotent. This result turns out to be useful to study the classes of PST-groups and PT-groups. Recall that a subgroup Hof a group Gis said to be S-permutable (or S-quasinormal, or π-quasinormal) in Gif HP =PH for all Sylow subgroups Pof G. It is clear that S-permutability is weaker than permutability and normality. According to a theorem of Kegel [10, Satz 1], every S-permutable subgroup is subnormal. S-permutability, like normality and permutability, is not a transitive relation. We say that a group Gis a PST-group if S-permutability is transitive in G, that is, if Ais an S-permutable subgroup of Band Bis an S-permutable subgroup of G, then Ais S-permutable in G. Applying Kegel’s theorem, PST-groups are exactly the groups in which every subnormal subgroup is S-permutable. This class contains the class of all groups in which normality is transitive (T-groups) and the class of all groups in which permutability is transitive (PT-groups). The last two classes have been widely studied ([1, 4, 6, 7, 10, 11, 15]). The structure of soluble PST-groups was obtained by Agrawal in [1]. It is proved there that a group Gis a soluble PST-group if and only if Ghas an abelian normal Hall subgroup of odd order Nsuch that G/N is nilpotent and the elements of Ginduce power automorphisms in N. In that result, if we force G/N to be a Dedekind group, we find Gasch¨utz’s characterisation of soluble T-groups ([7]), and if we impose that G/N is a nilpotent modular group, then we obtain Zacher’s characterisation of soluble PT-groups ([15]). The above results show that, in the soluble universe, the difference between these three classes is simply the Sylow structure. Our second result supports that claim and provides a unified viewpoint for the classes of PST,PT and T-groups in the general finite case. Theorem 2. Let Gbe a group. 2 1. Suppose that pis a prime number and that His an S-permutable psubgroup of G. If the Sylow p-subgroups of Gare modular (respectively, Dedekind), then His permutable (respectively, normal) in G. 2. Assume that His an S-permutable subgroup of G. If the Sylow subgroups of Gare modular (respectively, Dedekind), then His permutable (respectively, normal) in G. Taking this result into account, it seems natural to look for characterisations of the above classes in terms of the Sylow structure. This was done by Robinson ([11]) for the class of T-groups and by Beidleman, Brewster and Robinson ([4]) for the class of PT-groups. One of the purposes of this paper is to provide necessary and sufficient conditions on the Sylow structure for a group to be a soluble PST-group. As in the PT and T-cases, the procedure of defining local versions in order to simplify the study of the global properties has revealed itself as considerably useful. Since our approach depends heavily on a previous analysis of the classes of PT-groups and T-groups, the following definition needs to be stated. Definition 1. Let Gbe a group and pa prime. We say that G: 1. Enjoys property Cp(see [11]) if each subgroup of a Sylow p-subgroup Pof Gis normal in the normaliser NG(P). 2. Satisfies property Xp(as in [4]) if each subgroup of a Sylow p-subgroup Pof Gis permutable in the normaliser NG(P). Robinson ([11]) proved that a group Gis a soluble T-group if and only if Gsatisfies property Cpfor all primes pand, thirty-one years later, Beidleman, Brewster and Robinson proved that Gis a soluble PT-group if and only if Gsatisfies property Xpfor all primes p. These results would follow easily if one could prove that Cpand Xpare subgroup-closed. The subgroup-closed character of Cpfollows from the abnormality of the Sylow normalisers. Nevertheless, in the Beidleman, Brewster and Robinson approach, the subgroup-closed character of Xpfollows after an intensive study of the property Xpand its consequences for the group structure (see [4, Corollary 3]). In the following, we show that the subgroup-closed character of the property Xpfollows as a natural consequence of Theorem 1 and a new property called Yp, which can be considered as the “PST-version” of the properties Cpand Xp. Definition 2. Let pbe a prime number. A group Gis said to be a Yp-group when for all p-subgroups Hand Sof Gsuch that H≤S,His S-permutable in NG(S). 3 The above property can be compared to property Spintroduced by Beidleman and Heineken in [5]. Theorem 3. A group Gsatisfies Xp(respectively, Cp) if and only if Gsatisfies Ypand the Sylow p-subgroups of Gare modular (respectively, Dedekind). Since Ypis subgroup-closed, this result has the virtue of showing that the subgroup-closed character of Xpdepends exclusively on the modularity of the Sylow p-subgroups. It also shows that, in order to get a global characterisation of the soluble PST-groups, it is necessary to impose the subgroup-closed character in the definition Yp, as in the PST-case there are no restrictions on the Sylow p-subgroups. Assume that Gis a solubpe PST-group. If Hand Sare p-subgroups of Gsuch that H≤S, then His subnormal in NG(S). Now, by Agrawal’s Theorem, NG(S) is a PST-group. Therefore His S-permutable in NG(S). Consequently every soluble PST-group has property Yp. Our next result confirms that the converse is also true. Theorem 4. A group Gis a soluble PST-group if and only if Gsatisfies Yp for all primes p. Note that Theorem A of [4] is a consequence of Theorems 3 and 4. One of the main results of [4] is that a group Gsatisfies Xpif and only if Ghas modular Sylow p-subgroups and either Gis p-nilpotent or a Sylow p-subgroup Pof Gis abelian and Gsatisfies Cp. This result is a consequence of Theorem 3 and the following: Theorem 5. A group Gis a Yp-group if and only if Gis either p-nilpotent, or Ghas abelian Sylow p-subgroups and Gsatisfies Cp. Theorem C of [4] follows from Theorem 3 and Corollary 1. If pis the smallest prime divisor of the order of G, then Gis aYp-group if and only if Gis p-nilpotent. Theorem 5 has revealed itself to be useful to prove some interesting results on PST-groups. For instance, it is proved in [5, Theorem H] that a soluble group Gis a PST-group if and only if every subnormal subgroup permutes with every Carter subgroup of Gand the subnormal subgroups are hypercentrally embedded in G. As an application of Theorem 5, we prove in [3, Corollary 2] that the permutability with the Carter subgroups can be removed. Theorem 6 ([3]).A soluble group Gis a PST-group if and only if every subnormal subgroup of Gis hypercentrally embedded in G. 4 Another application of Theorem 5 is the following structure theorem for p-soluble groups with property Yp. Theorem 7 ([3]).Ap-soluble group has property Ypif and only if 1. either Gis p-nilpotent, or 2. G(p)/Op0G(p)is an abelian normal Sylow p-subgroup of G/Op0G(p) such that the elements of G/Op0G(p)induce power automorphisms in G(p)/Op0G(p). Here, G(p)denotes the p-nilpotent residual of G, that is, the smallest normal subgroup of Gsuch that G/G(p)is p-nilpotent. The paper is organised as follows. In Section 2 we study property Ypand its relation with the properties Cpand Xp. The local approach to the class of soluble PST-groups developed in [2] plays an important role. The proofs of the main results appear in Section 3. Finally, we give some non-soluble examples of groups with property Ypand a remark to show that any hope of creating a similar landscape out of the soluble universe which leads to a characterisation of PST,PT and T-groups is soon dispelled. 2 The property Yp In the sequel pwill be a fixed prime. Our first result confirms the subgroup-closed character of the property Cp. This is a consequence of the abnormality of the normalisers of the Sylow subgroups. Lemma 1. Cpis inherited by subgroups. Proof. Assume that Ghas the property Cpand let Ba subgroup of G. If Cis a Sylow p-subgroup of Band Dis contained in C, then Dis normal in NG(P) for every Sylow p-subgroup Pof Gcontaining C. Therefore if g∈NG(C), then Dis normal in hNG(P), NG(Pg−1)i. Since NG(P) is abnormal, it follows that g−1∈ hNG(P), NG(Pg−1)iand so g∈NG(D). Therefore Dis normal in NG(C) and Bhas property Cp. Bryce and Cossey ([6]) established local versions of some results of soluble T-groups. In particular, they characterised the soluble groups with the property Cpas the groups Gin which every p0-perfect subnormal subgroup of Gis normal in G. 5 Following Bryce and Cossey’s approach, it is proved in [2] that the soluble groups with property Xpare those whose p0-perfect subnormal subgroups are permutable with the Hall p0-subgroups and the Sylow p-subgroups are modular (see [2, Theorems 6 and 7]). Then the following definition arose: Definition 3 ([2]).We say that a group Gis a PSTp-group if Gis p-soluble and every p0-perfect subnormal subgroup is permutable with the Hall p0subgroups of G. According to [2, Theorem 8], a soluble group Gis a PST-group if and only if Gis a PSTp-group for all primes p. We say that a group G∈ U∗ pif it is p-soluble, and the p-chief factors of Gare cyclic groups and are G-isomorphic when regarded as G-groups by conjugation. In [2, Theorem 6] it is proved that a soluble group Gbelongs to PSTp if and only if G∈ U∗ p. The arguments used there still hold in the p-soluble universe. Therefore we have: Theorem 8. PSTp=U∗ p. In [2, Lemma 2] it is proved that the class of the PSTp-groups is quotientclosed. Theorem 8 shows that this class is also subgroup-closed. The characterisation of soluble PST-groups in terms of the Sylow structure follows from the following: Theorem 9. Ap-soluble group is a PSTp-group if and only if it satisfies Yp. We need the following elementary lemma. Lemma 2. Let Gbe a group. 1. If Ghas property Ypand Ais a normal p-subgroup of G, then G/A has property Yp. 2. If Ghas property Ypand Nis a normal p0-subgroup of G, then G/N has property Yp. Proof. 1. This follows immediately from the definition. 2. Assume that Ghas property Ypand let H/N ≤S/N be p-subgroups of G/N. Then there exist Sylow p-subgroups H1and S1of Hand S, respectively, such that H1is contained in S1and H=H1Nand S=S1N. Since Ghas Yp, it follows that H1is S-permutable in NG(S1). Therefore H/N =H1N/N is S-permutable in NG(S1)N/N = NG/N (S/N). This implies that G/N has Yp. 6 Proof of Theorem 9. Assume that Gsatisfies Yp. We prove that Gis a PSTpgroup by induction on |G|. Denote Op0(G) by Aand suppose that A6= 1. Let Hbe a p0-perfect subnormal subgroup of Gand let Bbe a Hall p0-subgroup of G. Then A≤Band B/A is a Hall p0-subgroup of G/A. Since G/A is aPSTp-group, it follows that HA/A permutes with B/A. Consequently H permutes with Band hence Gis a PSTp-group. Therefore we may assume that A=Op0(G) = 1. Let Nbe a minimal normal subgroup of G. Then Nis a p-group because Gis p-soluble. If N0is a subgroup of N, then N0is S-permutable in NG(N) = G. This means that if Qis a Sylow q-subgroup of Gfor q6=p, then N0is a Sylow p-subgroup of N0Qand so Qnormalises N0. Therefore Op(G) normalises every subgroup of N. Let Pbe a Sylow psubgroup of Gand let N1be a minimal normal subgroup of Pcontained in N. Then POp(G) = Gnormalises N1and so N1=N. This means that Nis cyclic of order p. By Lemma 2, we know that G/N has Yp. Therefore G/N is a PSTp-group by induction. Applying Theorem 8, we have that G/N is a U∗ p-group. In particular, G/N is p-supersoluble. Since Nis cyclic, it follows that Gis p-supersoluble. Then Ghas a normal Sylow p-subgroup Pcontaining the derived subgroup G0by [2, Lemma 1]. Let Hbe a p0-perfect subnormal subgroup of G. Then P∩His a normal Sylow p-subgroup of Hand so P∩H=Hsince His p0-perfect. Hence His a p-group and H≤P. Therefore His S-permutable in NG(P) = G. In particular, Hpermutes with the Hall p0-subgroups of G. Therefore Gis a PSTp-group. Conversely, suppose that Gis a PSTp-group. Suppose that Hand S are p-subgroups of Gsuch that H≤S. Then His a subnormal subgroup of NG(S), His p0-perfect and NG(S) is a PSTp-group because the class of PSTp-groups is subgroup-closed. Thus Hpermutes with every Hall p0subgroup Qof NG(S) and X=HQ is a subgroup of G. Then H≤Op(X) and Op(X) = HOp(X)∩Q=H. Therefore His normalised by Q. Consequently, OpNG(S)normalises Hand Ghas Yp. Note that every p-nilpotent group is U∗ p-group. Therefore by Theorem 8 and 9 we have: Corollary 2. If Gis p-nilpotent, then Ghas Yp. Another relevant property of groups with Ypis: Lemma 3. If Ghas Ypand if Pis a non-abelian Sylow p-subgroup of G, then NG(P)is p-nilpotent. 7 Proof. Let Hbe a subgroup of P. If Qis a Sylow q-subgroup of NG(P) for a prime p6=q, then HQ is a subgroup of G. This implies that His a subnormal Sylow p-subgroup of HQ and then Qnormalises H. Therefore every p0-element of NG(P) normalises every subgroup of P. Since Pis nonabelian, we can apply [8, Hilfssatz 5] to conclude that every p0-element of NG(P) actually centralises P. Consequently, NG(P) is p-nilpotent. Our proof of Theorem 5 depends on the relation between Ypand pnormality. Recall that if pis a prime, a group Gis said to be p-normal if it satisfies the following property: If Pis a Sylow p-subgroup of Gand Z(P) is contained in Pgfor some g∈G, then Z(P) = Z(Pg). This property is closely related to property Yp. In fact, we have: Lemma 4. If Gsatisfies Yp, then Gis p-normal. Proof. Suppose that Gsatisfies Yp. Let Pbe a Sylow p-subgroup of Gand let gbe an element of Gsuch that Z=Z(P)≤Pg. Suppose that Zis not a normal subgroup of Pg. Then (see Burnside’s Theorem, [9, Satz IV.5.1]) there exists an element g∈Gof order qbfor a prime q6=psuch that g /∈NG(Z), J=ZZg· · · Zgqb−1is a p-group and g∈NG(J)\CG(J). But gis ap0-element of NG(J) and Gis a Yp-group. Consequently ginduces a power automorphism on J. In particular, we get the contradiction g∈NG(Z). Therefore Z(P) is a normal subgroup of Pg. Then Z(Pg−1) = Z(P)g−1 is a normal subgroup of P. By [9, Hilfssatz IV.5.2], since Z(P) is a characteristic subgroup of P, we have that Z(P) = Z(Pg−1) and Z(P) = Z(Pg). That proves that Gis p-normal. 3 Proofs of the main results The next result is the p-soluble version of Theorem 1. Lemma 5. Let pbe a prime. Assume that Gis a p-soluble group with modular Sylow p-subgroups. If Pis a Sylow p-subgroup of Gsuch that NG(P) is p-nilpotent, then Gis p-nilpotent. Proof. Assume the result is false and let Gbe a counterexample of least order. Then for each non-trivial normal subgroup Nof G, it follows that G/N is p-nilpotent. Therefore, since the class of p-nilpotent groups is a saturated 8