On a class of p-soluble groups

This is an electronic version of an article published as Algebra Colloquium 12(2):263–267 (2005) DOI: 10.1142/S1005386705000258. Copyright World Scientific Publishing Company http://www.worldscientific.com/doi/abs/10.1142/S1005386705000258 On a class of p-soluble groups∗ A. Ballester-Bolinches Departament d’Àlgebra, Universitat de València Doctor Moliner 50, 46100 Burjassot (València), Spain e-mail: A[email protected] R. Esteban-Romero and M. C. Pedraza-Aguilera Departament de Matemàtica Aplicada, Universitat Politècnica de València Camí de Vera, s/n, 46022 València (Spain) e-mails: [email protected]v.es and mp[email protected]v.es Abstract Let pbe a prime. The class of all p-soluble groups Gsuch that every p-chief factor of Gis cyclic and all p-chief factors of Gare Gisomorphic is studied in this paper. Some results on T-, PT -, and PST -groups are also obtained. Keywords: finite groups, permutability, subnormality. 1991 Mathematics Subject Classification: 20D20, 20D35. 1 Introduction All groups considered in the paper will be finite. Let pbe a prime number. Denote by U∗ pthe class composed of all p-soluble groups Gsuch that every p-chief factor of Gis cyclic (Gis p-supersoluble) and all p-chief factors of Gare G-isomorphic. For a p-soluble group G, the following statements are pairwise equivalent: 1. Gbelongs to U∗ p. 2. Every p0-perfect subnormal subgroup of Gpermutes with every Hall p0-subgroup of G([2, Theorem 6]). ∗Supported by Grant BFM2001-1667-C03-03, MCyT (Spain) and FEDER (European Union) 1 3. If H≤Kare p-subgroups of G, then Hpermutes with all Sylow subgroups of NG(K)([5, Theorem 9]). 4. Gis either p-nilpotent, or Ghas an abelian Sylow p-subgroup Pand every subgroup of Pis normal in NG(P)([5, Theorem 5]). 5. Either Gis 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), where G(p)denotes the p-nilpotent residual of G, that is, the smallest normal subgroup of Gsuch that G/G(p)is p-nilpotent ([4, Theorem A]). These characterisations let the class U∗ pplay a major role in the study of three interesting classes of groups: T-groups (or groups in which normality is transitive), PT -groups (or groups in which permutability is transitive), and PST -groups (or groups in which permutability with Sylow subgroups is transitive). These classes have been widely studied (see [1, 2, 4, 5, 7, 8, 9, 10, 13]). The main goal of this paper is to study the behaviour of U∗ pas a class of groups and apply the results to get information about the classes of T-, PT -, and PST -groups. It is clear that U∗ pis a subgroup-closed homomorph. However, the direct product of a symmetric group of degree 3with a cyclic group of order 3shows that it is not closed under taking direct products. In particular, U∗ pis not a formation. More precisely, we have: Theorem A. The class of all p-nilpotent groups is the largest formation contained in U∗ p. Since a soluble PST -group is a U∗ p-group for all primes p([2, Theorems 6 and 8]), we have: Corollary 1. The class of all nilpotent groups is the largest formation contained in the class of all soluble PST -groups. The class U∗ pis not saturated in general (see [3] or [12]). However we have: Theorem B. Let Gbe a group. The following statements are equivalent: 1. for every subgroup Hof G,H/Φ(H)is a U∗ p-group, and 2. Gis a U∗ p-group. 2 This theorem is a consequence of the following result, which is proved in [6]: Theorem C. Let Gbe a p-supersoluble group. The following statements are equivalent: 1. Gbelongs to U∗ p. 2. Gdoes not have any subgroup of the form X= [P]Q, where pand q are primes such that qf|p−1, with f≥1,iis a primitive root of unity modulo p,j= 1 + kqf−1, with 0< k < q,P=ha, biis an elementary abelian group of order p2,Q=hziis a cyclic q-group of order qrwith r≥fsuch that az=ai,bz=bij. Following Van der Waall and Fransman [12], we say that a group Gis a T0-group (respectively, a PT 0-group, a PST 0-group) if G/Φ(G)is a T-group (respectively, a PT -group, a PST -group). As a consequence of Theorem B, [2, Theorems 6 and 8] and the fact that the class of soluble PST -groups is subgroup-closed , we have the following result. Corollary 2. Let Gbe a group. The following statements are equivalent: 1. Every subgroup of Gis a PST 0-group. 2. Every subgroup of Gis a PST -group. 3. Gis a soluble PST -group. Assume now that every subgroup of Gis T0-group. By Corollary 2, G is a soluble PST -group. Applying Agrawal’s theorem [1], Ghas an abelian normal Hall subgroup Dof odd order complemented by a nilpotent subgroup Bsuch that every subgroup of Dis normal in G. Consequently, B0centralises every subgroup of D. In fact we have: Theorem D. Let Gbe a group. The following statements are pairwise equivalent: 1. Gis a soluble PST -group, 2. Gis supersoluble and has a normal abelian subgroup Dof odd order and a nilpotent subgroup Bsuch that G=DB, with gcd(|D|,|B|) = 1, B0EGand G/B0is a T-group, and 3 3. Every subgroup of Gis a T0-group. 4. Every subgroup of Gis a PT 0-group. Note that Van der Waall and Fransman’s theorem [12, Theorem 3.10] is the equivalence between 2 and 3. Applying these results, we are able to give an alternative proof of the main result of [3]. We will also use the following result, which is a particular case of a theorem proved in [6]. Theorem E. Let Gbe a p-soluble group such that all proper subgroups of G belong to U∗ p, but Gitself does not belong to U∗ p. Then Ghas a normal Sylow p-subgroup Pwhich is complemented by a non-normal cyclic subgroup whose order is a power of a prime q6=p. Theorem F. Assume that every proper subgroup of a group Gis a T0-group, but Gitself is not a T0-group. Then: 1. G=PQ, where Pis a Sylow p-subgroup of Gand Qis a Sylow qsubgroup of Gfor some distinct primes pand q; 2. P / G and Qis a non-normal cyclic subgroup of G; 3. G/Φ(G)is a minimal non-T-group. 2 Proofs Proof of Theorem A. Let Fbe a formation contained in the class U∗ p. Assume that Gis a group such that G∈F, but Gis not p-nilpotent. Given a group X, let us denote by X(p)the p-nilpotent residual of X. Since Fis a formation, we have that H=G/Op0G(p)belongs to F. By [4, Theorem A], H(p)is an abelian normal Sylow p-subgroup of Hsuch that the elements of Hinduce power automorphisms in H(p). Since His not p-nilpotent, there exists a p0-element x∈Hsuch that xdoes not centralise H(p). On the other hand, since Fis a formation, we have that H×H∈F. Moreover H×Hdoes not belong to U∗ p, because (x, 1) centralises the p-chief factors of the second factor of the direct product, but does not centralise the p-chief factors of the first factor, a contradiction. Proof of Theorem B. It is clear that if Gis a U∗ p-group, then for every H≤G, His a U∗ p-group and hence H/Φ(H)is a U∗ p-group. Assume that the converse is false. Let Gbe a group of minimal order such that for every subgroup Hof G,H/Φ(H)is a U∗ p-group, but Gitself is 4 not a U∗ p-group. By minimality of G, we have that all proper subgroups H of Gbelong to U∗ p, but Gitself does not. By Theorem C, it follows that G has the form G= [P]Q, where pand qare primes such that qf|p−1, with f≥1,iis a primitive root of unity modulo p,j= 1+ kqf−1, with 0< k < q, P=ha, biis an elementary abelian group of order p2,Q=hziis a cyclic q-group of order qrwith r≥fsuch that az=ai,bz=bij. Since haiand hbiare normal subgroups of G, it follows that haiQand hbiQare maximal subgroups of G. Hence Φ(G)≤ haiQ∩ hbiQ=Q. Therefore G/Φ(G)can be expressed as a semidirect product of an elementary abelian normal subgroup h¯a,¯ biof order p2by a cyclic subgroup h¯zisuch that ¯a¯z= ¯aiand ¯ b¯z= ¯aij. Since G/Φ(G)satisfies U∗ p, it follows that habiis normalised by z. Hence aibij belongs to habi, which implies that ij≡i(mod qf). Thus j≡1 (mod qf), but j= 1 + kqf−1with 0< k < q, a contradiction. Hence Gbelongs to U∗ p. Proof of Theorem D. (1) implies (2) Assume that Gis a soluble PST -group. By Agrawal’s theorem [1], there exists an abelian Hall subgroup Dof odd order complemented by a nilpotent subgroup Bsuch that every subgroup of Dis normal in G. Let d∈D. Since hdiis a normal subgroup of G, it follows that G/CG(hdi) is abelian. Hence B0≤CG(hdi). It follows that B0≤CG(D). Consequently, B0is a normal subgroup of G. Since Gis a soluble PST -group, we have that G/B0is a PST -group. Moreover, all Sylow subgroups of G/B0are abelian, because they are Sylow subgroups of the abelian group Dor isomorphic to Sylow subgroups of the abelian group B/B0. Therefore, G/B0is a T-group by [5, Theorem 2]. (2) implies (3) Assume that Gis a supersoluble group with an abelian normal Hall subgroup Dof odd order complemented by a nilpotent subgroup Bsuch that B0is normal in Gand G/B0is a T-group. Since Bis nilpotent, we have that B0≤Φ(B). Hence B0≤Φ(G), because B0is a normal subgroup of B. In particular, G/Φ(G)is a soluble T-group. Let Hbe a subgroup of G. Since Gis a soluble PST -group, we have that His a soluble PST -group. Hence the nilpotent residual HNof His a normal Hall subgroup of Hof odd order complemented by a subgroup HBwhich can be assumed to be contained in B. Since (HB)0≤B0and (HB)0≤Φ(HB), because HBis nilpotent, and (HB)0is a normal subgroup of H, we obtain that (HB)0≤Φ(H). Hence H/Φ(H)is again a soluble T-group. Hence Gis aT0-group. It is clear that (3) implies (4). Assume now that every subgroup of Gis aPT 0-group. Then every subgroup of Gis a PST 0-group. By Corollary 2, it follows that Gis a soluble PST -group. Therefore (4) implies (1) and the 5 circle of implications is complete. Proof of Theorem F. Assume that Gis a minimal non-T0-group. Then all proper subgroups of Gare T0-groups. From Theorem D, we have that all proper subgroups of Gare soluble PST -groups. In particular, all proper subgroups of Gare supersoluble, which implies that Gitself is soluble (see [11]). If Gwere a PST -group, then Gwould be a T0-group by Theorem E, a contradiction. Therefore Gis not a PST -group. Hence there exists a prime p such that Gis a minimal non-U∗ p-group. By Theorem E, Gcan be expressed as G=PQ satisfying conditions 1 and 2. Since Φ(P)≤Φ(G)because Pis a normal subgroup of G, it follows that G/Φ(G)has a normal abelian Sylow p-subgroup PΦ(G)/Φ(G). Consequently, G/Φ(G)has abelian Sylow subgroups. 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