Sylow permutable subnormal subgroups of finite groups II

Sylow permutable subnormal subgroups of finite groups II∗ 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 30th March 2001 Abstract In this paper a local version of Agrawal’s theorem about the structure of finite groups in which Sylow permutability is transitive is given. The result is used to obtain new characterisations of this class of finite groups. AMS subject classification: 20D30, 20E15. ∗Supported by Proyecto PB97-0674-C02-02 and Proyecto PB97-0604 from DGICYT, Ministerio de Educaci´on y Ciencia 1 1 Introduction Throughout the paper, the word group means finite group. A subgroup Hof a group Gis said to be S-permutable in Gif Hpermutes with every Sylow subgroup of G. By the results of Kegel [9] and Schmid [12], every S-permutable subgroup is subnormal and the set of all S-permutable subgroups is a sublattice of the lattice of all subnormal subgroups of G. We say that a group Gis a PST-group when the above lattices coincide, that is, when every subnormal subgroup is S-permutable. It follows that a group Gis a PST-group exactly when the S-permutability is a transitive relation. 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. Soluble PST-groups were first studied by Agrawal [1] in 1975, and recently by Alejandre, the first author and Pedraza-Aguilera [2], the authors [3], and Beidleman and Heineken [5]. Soluble PT-groups have been investigated by Zacher [13] in 1964, and more recently by Beidleman, Brewster and Robinson [4]. T-groups have been widely studied [6, 8, 10, 11]. The paper [3] provides a unified viewpoint for the classes of soluble PST, PT and T-groups in terms of their Sylow structure. The approach we have been following started in a paper of Bryce and Cossey [6], where a local version of some of the results on T-groups was established. This approach was also followed in the papers [2, 5]. One of our purposes in this paper is to give a local version of the classical theorems of Gasch¨utz, Zacher and Agrawal by using the results of [3]. We also provide new local characterisations of the soluble PST-groups motivated by the results of the recent paper by Beidleman and Heineken [5]. Let us start with the classical characterisation of soluble PST-groups. Theorem 1 (Agrawal). A group Gis a soluble PST-group if and only if Ghas an abelian normal Hall subgroup Nof odd order such that G/N is nilpotent and the elements of Ginduce power automorphisms in N. In that result, if we add ‘G/N nilpotent modular group,’ we obtain Zacher’s characterisation of soluble PT-groups [13], and if we put ‘G/N Dedekind,’ we get Gasch¨utz’s characterisation of soluble T-groups [8]. The above results indicate that the three classes are closely related. However, there is a very important difference between them: the Sylow structure. In the case of T-groups, the Sylow subgroups are Dedekind, and modular in the case of PT-groups. There are no restrictions on Sylow subgroups in the PST-case because every nilpotent group is a PST-group. 2 For a prime p, Bryce and Cossey [6] introduced the class Tpof all soluble groups Gfor which every subnormal p0-perfect subgroup of Gis normal. They prove: Theorem 2. A soluble group is a T-group if and only if it is a Tp-group for all primes p. Motivated by Theorem 2, Alejandre, the first author and Pedraza-Aguilera introduced in [2] the class PSTpof all soluble groups Gin which every subnormal p0-perfect subgroup in Gpermutes with every Hall p0-subgroup of G. They have proved: Theorem 3. A soluble group Gis a PST-group if and only if Gis a PSTpgroup for all primes p. As we have noted in [3], Theorem 3 holds equally if the word ‘soluble’ is replaced by ‘p-soluble’ in the definition of the class of PSTp-groups. Note that [2, Theorem 6] also holds in the p-soluble universe. Therefore we have: Theorem 4. Ap-soluble group Gis a PSTp-group if and only if Gis p-supersoluble and all its chief p-factors are G-isomorphic regarded as Gmodules by conjugation. In the paper [5], Beidleman and Heineken define, for a prime p, the class Tp (respectively, T00 p) of all soluble groups Gfor which every p0-perfect subnormal subgroup of Gis permutable (respectively, S-permutable) and prove: Theorem 5. Let Gbe a soluble group. Then 1. Gis a PT-group if and only if it is a T0 p-group for all primes p. 2. Gis a PST-group if and only if it is a T00 p-group for all primes p. Combining Theorem 4 and Theorem 5 of [3], we have: Theorem 6. Ap-soluble group Gis a PSTp-group if and only if Gis either p-nilpotent, or Ghas an abelian Sylow p-subgroup Pand every subgroup of Pis normal in NG(P). Using Theorem 6, we prove in this paper a structure theorem for PSTpgroups which can be considered as the local version of Agrawal’s theorem. Before beginning the presentation of that theorem, the reader should be aware of the notation that is used. Let pbe a prime and let Ep0Spbe the saturated formation of all pnilpotent groups. For each group G, we denote G(p) the Ep0Sp-residual of G, that is, the smallest normal subgroup of Gsuch that G/G(p) is p-nilpotent. 3 Theorem A. A group Gis a PSTp-group if 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). A local version of Gasch¨utz’s theorem is obtained by adding in 1 in Theorem A the condition ‘Ghas Dedekind Sylow p-subgroups.’ A local version of Zacher’s theorem is obtained by adding in 1 in Theorem A the condition ‘Ghas modular Sylow p-subgroups.’ The result is used in the paper to give a succinct proof of Agrawal’s theorem. Let pbe a prime. In any group Gthere is a unique maximum normal p-nilpotently embedded subgroup, denoted here by Zp(G). Hence Zp(G) is pnilpotent and there is a G-invariant series in Zp(G) such that the chief factors whose order is a power of pare central, while G/Zp(G) has no central normal p-subgroups. Zp(G) is the Ep0Sp-hypercentre of G([7, IV,6.8]), and it is contained in Z∞(G), the hypercentre of G. As a consequence of Theorem A we have: Corollary 1. Let Gbe a PSTp-group. Then ZpG/Op0G(p)is a Hall p0subgroup of Op0,p(G)/Op0G(p)and Op0,p(G)/Op0G(p)=G(p)/Op0G(p)× ZpOp0G(p). We say that a subgroup Hof Gis p-hypercentrally embedded (respectively, hypercentrally embedded) if H/ CoreG(H)≤ZpG/ CoreG(H)(respectively, H/ CoreG(H)≤Z∞G/ CoreG(H)). Beidleman and Heineken [5] proved 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. We prove in the following that the permutability with the Carter subgroups can be removed. Theorem B. Ap-soluble group Gis a PSTp-group if and only if every p0-perfect subnormal subgroup of Gis p-hypercentrally embedded in G. Corollary 2. A soluble group Gis a PST-group if and only if every subnormal subgroup of Gis hypercentrally embedded in G. 4 2 Proofs Proof of Theorem A. Let Gbe a PSTp-group and assume that Gis not pnilpotent. Then G(p)6= 1 and pdivides |G(p)|. Applying Theorem 6, Ghas an abelian Sylow p-subgroup Psuch that every subgroup of Pis normal in NG(P). Since G/Op0G(p)inherits the hypotheses of G, we can assume without loss of generality that Op0G(p)= 1. G(p) is contained in G0, the commutator subgroup of G, which is p-nilpotent because Gis p-supersoluble. Hence G(p) is p-nilpotent and so G(p) is a p-group contained in P. In particular, G(p) is abelian. Note that [7, IV,5.18] still holds in the p-soluble universe. Consequently G(p) is complemented in Gby every p-nilpotent projector. Let Dbe one of them and assume that pdivides |D|. Then there exists a p-chief factor A/B of Gsuch that G(p)≤B < A. This implies that A/B is central in G. By Theorem 4, all p-chief factors of Gare G-isomorphic. In particular, every chief factor below G(p) is central, a contradiction. Consequently Dis a Hall p0-subgroup of Gand G(p) is a normal Sylow p-subgroup of G. Hence every subgroup of G(p) is normal in NGG(p)=G. Conversely, suppose that Gis either p-nilpotent or 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). If Gis p-nilpotent, then Gis p-supersoluble and all p-chief factors are central. Hence Gis a PSTp-group by Theorem 4. Assume now that G(p)/Op0G(p)is an abelian Sylow p-subgroup of the group G/Op0G(p)and every subgroup of G(p)/Op0G(p)is normal in G/Op0G(p). Let Hbe a subnormal p0-perfect subgroup of Gand let Dbe a Hall p0-subgroup of G. Denote T=HOp0(G). Then T=T∩G(p)(T∩D) and T∩G(p)/Op0G(p)is normal in G/Op0G(p). Therefore T∩G(p) is normal in G. Now HD =TD =T∩G(p)D, which is a subgroup of G. Consequently Hpermutes with Dand Gis a PSTp-group. Proof of Corollary 1. It is clear that we may assume that G(p)6= 1 and Op0G(p)= 1. By Theorem A, G(p) is an abelian normal Sylow p-subgroup of Gand so G(p)≤Op0,p(G). Let Dbe a Hall p0-subgroup of G. Then G=G(p)D. Therefore Op0,p(G) = G(p)D∩Op0,p(G). Since Op0,p(G) is p-nilpotent and D∩Op0,p(G) is a Hall p0-subgroup of Op0,p(G), it follows that D∩Op0,p(G) is normal in G. Consequently, D∩Op0,p(G)≤CGG(p), which coincides with Zp(G) by [7, IV,6.14]. On the other hand, since G(p) is abelian, then Dis actually a p-nilpotent projector of Gby [7, IV,5.18]. Since, by [7, IV,6.14], Zp(G)≤D, it follows that D∩Op0,p(G) = Zp(G). The 5 proof of the corollary is now complete. Proof of Agrawal’s Theorem (Theorem 1). Suppose that Gis a PST-group. Let πbe the set of all primes psuch that Ghas a non-central chief p-factor. Then T=GN=hG(p)|pis a primei=hG(p)|p∈πi, because G(p) = 1 if pdoes not belong to π. We prove that Tis a Hall πsubgroup of G. Since Gis supersoluble, we have that G0is nilpotent. Hence Tis nilpotent. Assume that qis a prime dividing the order of Tsuch that q /∈π. Let Tq0be a Hall q0-subgroup of T. Then Tq0is a normal subgroup of G. Since q /∈π, we have that all chief q-factors of Gare central. Therefore G/Tq0is nilpotent. Moreover, Tq0is a normal subgroup of G, and G/Tq0is nilpotent. This implies that T=Tq0, and Tis a q0-group, a contradiction. Consequently Tis a π-subgroup of G. If pbelongs to π, then G(p) contains an abelian Sylow p-subgroup of Gby Theorem A. Hence Tis abelian. Moreover, 2 cannot belong to π, because a PST2-group is always 2-nilpotent. We must show now that every subgroup of Tis normal in G. Since Tis nilpotent, we may assume without loss of generality that Tis a p-group for some prime p. Then Op0(T) = Op0G(p)and G(p) is a Sylow p-subgroup of G. Hence G(p) = Tand every subgroup of Tis normal in NG(T) by Theorem A. Conversely, suppose that N=GNis an abelian normal Hall subgroup N of odd order of Gsuch that every subgroup of Nis normal in G. Then it is clear that Gsatisfies the conditions of Theorem A for all primes p. Therefore Gis a PSTp-group for all primes pand then Gis a PST-group. The following lemma is needed to prove Theorem B. Lemma 1. If Ris a Hall p0-subgroup of a group G, then Zp(G)≤NG(R). Proof. Let Gbe a counterexample of minimal order to this lemma. Let R be a Hall p0-subgroup of Gsuch that Zp(G)6≤ NG(R). Let T=Zp(G)R. If Tis a proper subgroup of G, then Zp(G)≤Zp(T), which is contained in NT(R) by the minimal choice of G, a contradiction. Therefore T=G and G=Zp(G)R. Since Ris a p0-group, Ris p-nilpotent. Let Xbe a maximal p-nilpotent subgroup of Gcontaining R. By [7, IV,6.14], it follows that Zp(G)≤X. Consequently G=Xand Gis p-nilpotent. This implies that Ris a normal subgroup of G, final contradiction. Proof of Theorem B. Suppose that Gis a PSTp-group. Then, by Theorem 6, either Gis p-nilpotent, or Ghas an abelian Sylow p-subgroup Pand every subgroup of Pis normalised by the normaliser NG(P). 6 In the first case, if Gis p-nilpotent, every p0-perfect subnormal subgroup is p-hypercentrally embedded. On the other hand, assume that a Sylow psubgroup Pis abelian and every subgroup of Pis normal in NG(P). Note that the proof of [6, Theorem 2.3] works in the p-soluble universe. Hence in this case every p0-perfect subnormal subgroup of Gis normal. Consequently HG=Hand H/HG≤Zp(G/HG). Conversely, suppose that every p0-perfect subnormal subgroup Hof Gis p-hypercentrally embedded. Let Hbe a subnormal p0-perfect subgroup of G, and let Rbe a Hall p0-subgroup of G. Since H/HG≤Zp(G/HG) and RHG/HGis a Hall p0-subgroup of G, it follows by Lemma 1 that H/HG normalises RHG/HG. In particular, Hpermutes with RHG. This implies that Hpermutes with R. Therefore Gis a PSTp-group. Proof of Corollary 2. Suppose that Gis a PST-group of minimal order with a subnormal subgroup Hnot hypercentrally embedded in G. Assume that His of minimal order. We can clearly assume that HG= 1. Hence H is nilpotent by [12, Proposition A] and every Sylow subgroup of His Spermutable in Gby [12, Proposition B]. From the minimality of H, we can suppose that His a p-group for a prime p. Since Gis a PSTp-group, H is p-hypercentrally embedded in Gby Theorem B. Since His a subnormal p-subgroup, its normal closure K=hHGiis a p-group. Let A/B be a chief factor of Gsuch that B < A ≤K. Since B/A is a p-group and K≤Zp(G), it follows that B/A is central in G. Thus Kis contained in Z∞(G). In particular, His hypercentrally embedded in G. Conversely, suppose that every subnormal subgroup of Gis hypercentrally embedded in G. Let pbe a prime number and let Hbe a p0-perfect subnormal subgroup. Since His hypercentrally embedded in G, it follows that His phypercentrally embedded in G. Hence Gis a PSTp-group by Theorem B. Therefore Gis a PST-group by Theorem 3. 3 Examples Example 1.The result of Corollary 1 cannot be sharpened to Op0,p(G) = G(p)Zp(G) when Op0G(p)6= 1. The symmetric group Σ3of degree 3 has an irreducible and faithful module over the field of 2 elements. Let V=hv1, v2i and W=hw1, w2ibe two Σ3-isomorphic copies of these modules. There is an action of the cyclic group C2=hziof order 2 on V×Wsuch that vz j=wj, wz j=vjfor j∈ {1,2}. Hence we can consider the semidirect product G= [V×W](Σ3×C2) with respect to this action. This group is clearly aPST3-group, because its Sylow 3-subgroup has order 3, but Z3(G) = 1, 7 O30,3(G) = (V×W)(A3×C2) and G(3) = (V×W)A3, whence O30,3(G)6= G(3)Z3(G). Example 2.One might ask whether a p-soluble group Gis a PSTp-group if and only if every p0-perfect subnormal subgroup of Gis hypercentrally embedded. The answer is negative: The dihedral group D8of order 8 has an irreducible and faithful module of dimension 2 over the field of 3 elements. Let Gbe the corresponding semidirect product. This group is a PST2-group, because it is 2-nilpotent. On the other hand, Gpossesses 20-perfect core-free subnormal subgroups of order 6 isomorphic to the symmetric group Σ3of degree 3. These subgroups cannot be hypercentrally embedded, because they are not nilpotent. This example also shows that a PSTp-group does not have property T00 p in general. Example 3.There are non-PSTp-groups such that every subnormal p-subgroup is hypercentrally embedded. For instance, let G=hxi × Σ4, where hxiis a cyclic group of order 3 and Σ4is the symmetric group of degree 4. The subnormal 3-subgroups of Gare 1 and hxi. Both are hypercentrally embedded in G, because they are normal. But Gis not a PST3-group, because the 3-factor hxiis central but A4/V4is not. Example 4.Even in the case that all subnormal p-subgroups of Gare hypercentrally embedded for all pwe cannot ensure that a soluble group is a PST-group: The cyclic group C3of order 3 has an irreducible and faithful module V7over the field of 7 elements. The group C3×Σ3acts on V7with kernel Σ3. Let Gbe the semidirect product corresponding to this action. The unique subnormal subgroups of prime-power order of Gare 1, A3and V7. 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