Products of formations of finite groups

Document downloaded from: This paper must be cited as: The final publication is available at Copyright Additional Information http://hdl.handle.net/10251/19042 Ballester Bolinches, A.; Calvo Lopez, C.; Esteban Romero, R. (2006). Products of formations of finite groups. Journal of Algebra. 2(299):602-615. doi:10.1016/j.jalgebra.2006.01.003 http://dx.doi.org/10.1016/j.jalgebra.2006.01.003 Elsevier This paper has been published in Journal of Algebra, 299(2):602-615 (2006). Copyright 2006 by Elsevier. http://dx.doi.org/10.1016/j.jalgebra.2006.01.003 This paper has been published in Journal of Algebra, 299(2):602–615 (2006). Copyright 2006 by Elsevier. The final publication is available at www.sciencedirect.com. http://dx.doi.org/10.1016/j.jalgebra.2006.01.003 http://www.sciencedirect.com/science/article/pii/S0021869306000068 Products of formations of finite groups A. Ballester-Bolinches ∗ Departament d’ ` Algebra, Universitat de Val`encia, Dr. Moliner, 50, E-46100 Burjassot, Val`encia, Spain. Clara Calvo Departament d’ ` Algebra, Universitat de Val`encia, Dr. Moliner, 50, E-46100 Burjassot, Val`encia, Spain. R. Esteban-Romero Departament de Matem`atica Aplicada-IMPA, Universitat Polit`ecnica de Val`encia, Cam´ı de Vera, s/n, E-46022 Val`encia, Spain Abstract In this paper criteria for a product of formations to be X-local, Xa class of simple groups, are obtained. Some classical results on products of saturated formations appear as particular cases. Key words: Finite groups, X-local formation, ω-local formation, formation products 1991 MSC: 20D10, 20F17 1 Introduction All groups considered in this paper are tacitly assumed to be finite. Recall that a formation Fis a class of groups which is closed under taking homomorphic images and subdirect products. The second condition ensures the existence of the F-residual UFof each group U, that is, the smallest normal ∗Corresponding author. Email addresses: [email protected] (A. Ballester-Bolinches), [email protected] (Clara Calvo), [email protected] (R. Esteban-Romero). Preprint submitted to Elsevier Science 15 December 2005 subgroup of Uwhose factor group is in F. A formation Fis said to be saturated if U∈Fwhenever the Frattini factor group U/Φ(U) is in F. Gasch¨utz introduced the concept of local formation, which allows us to construct saturated formations. In fact, the Gasch¨utz-Lubeseder-Schmid theorem states that the family of local formations coincides with the one of saturated formations (see [1, Section IV] for details). It was proved by Gasch¨utz and Lubeseder in the soluble universe, and by Schmid in the general one. Baer gave an alternative generalization of the theorem of Gasch¨utz and Lubeseder to the general universe. He studied a different concept of local formation, using the simple components of a chief factor to label it, rather than the primes dividing its order (see [1, Section IV]). He found a family of formations, the Baer-local formations, which contains the family of local ones and coincides with it in the soluble universe. Baer-local formations are exactly the ones that are closed under extensions through the Frattini subgroup of the soluble radical, i. e., the solubly saturated ones. This is known as Baer’s theorem (see [1, IV, 4.17]). Another approach to the Gasch¨utz-Lubeseder theorem in the finite universe is due to Shemetkov. He uses functions assigning a certain formation to each group (he recently calls them satellites) and introduces the notion of composition formation (see the recent survey [2]). It turns out that composition formations are exactly the Baer-local ones. F¨orster introduced in [3] the concept of X-local formation, where Xis a class of simple groups with a completeness property. The main aim of his work was to present a common extension of the Gasch¨utz-Lubeseder-Schmid and Baer theorems. F¨orster also defined a Frattini-like subgroup Φ∗ X(U) for each group U, which enables him to introduce the concept of X-saturation. He proves that X-saturated formations are exactly the X-local ones. In [4], an alternative XFrattini subgroup ΦX(U) of a group Uis introduced. Except for a particular case, it is defined as ΦOX(U), where OX(U) is the largest normal subgroup of Uwhose composition factors belong to X. It was proved in [4] that the new X-saturated formations also coincide with the X-local ones. A local approach to the saturation is the ω-saturation, where ωis a nonempty set of primes. We say that a formation Fis ω-saturated if the condition U/Φ(U)∩Oω(U)∈Falways implies that U∈F. This concept arises spontaneously when the saturation of formation products is considered (see [5]). Given two classes Yand Zof groups, a product class can be defined by setting YZ = (U∈E|there is a normal subgroup Vof U such that V∈Yand U/V ∈Z), However this class is not in general a formation when Yand Zare formations. But there is a way of modifying the above definition to ensure that the class 2 product of two formations is again a formation. If Fand Gare formations, the formation product or Gasch¨utz product of Fand Gis the class F◦Gdefined by F◦G:= (U∈E|UG∈F). F◦Gis again a formation and if Fis closed under taking subnormal subgroups, then FG =F◦G(see [1, IV, 1.7 and 1.8]). It is well known that the formation product of two local formations is again a local formation (see [1, IV, 3.13 and 4.8]). However, the formation product of two X-local formations is not in general an X-local formation, as it is shown in Example 6. Taking this into account, the following question arises: Which are the precise conditions on two X-local formations Fand Gto ensure that F◦Gis an X-local formation? This question was studied by Salomon in [6] for Baer-local formations. We present a complete answer in Section 3. We prove that the formation product of a local formation and an X-local one is X-local. In particular, [1, IV, 3.13 and 4.8] follow from our results. In Section 4, which is independent of Section 3, we study when the product of two arbitrary formations Fand Gis X-local. On the other hand, Shemetkov posed the following question in The Kourovka Notebook ([7]): Question 10.72 (Shemetkov). To prove indecomposability of Sp,pa prime, into a product of two non-trivial subformations. This question was solved positively by Shemetkov and Skiba in [8]. In Section 5 we deal with ω-saturated formations and we prove a general version of this conjecture as a corollary of a more general result. 2 Preliminaries We begin with the concept of X-local formation. It was introduced by F¨orster in [3]. Let Jdenote the class of all simple groups. For any subclass Yof J, we write Y0:= J\Y. Denote by EYthe class of groups whose composition factors belong to Y. A chief factor which belongs to EYis called a Y-chief factor. If pis a prime, we write Ypto denote the class of all simple groups S∈Y such that p∈π(S). In the sequel it will be convenient to identify the prime p with the cyclic group Cpof order p. The class of all simple abelian groups is 3 denoted by P. The class of all π-groups, where πis a set of primes, is denoted by Eπ. If π={p}, then Eπ=Sp. Throughout this paper, Xdenotes a fixed class of simple groups satisfying that π(X) = char(X), where π(X) := {p∈P|there exists S∈Xsuch that pdivides |S|} and char(X) := {p∈P|Cp∈X}. Definition 1 ([3]). An X-formation function fassociates with each X∈ char(X)∪X0a formation f(X) (possibly empty). If fis an X-formation function, then LFX(f) is the class of all groups Usatisfying the following two conditions: (1) If V/W is an Xp-chief factor of U, then U/ CU(V/W)∈f(p). (2) If U/L is a monolithic quotient of Usuch that Soc(U/L) is an X0-chief factor of U, then U/L ∈f(E), where Eis the composition factor of Soc(U/L). The class LFX(f) is a formation ([3]). A formation Fis said to be X-local if there exists an X-formation function fsuch that F= LFX(f). In this case we say that fis an X-local definition of For that fdefines F. The X-formation function fis full if Spf(p) = f(p) for every p∈char(X) and fis integrated if f(S)⊆Ffor every simple group S∈char(X)∪X0. Examples 2. (1) Each formation Fis X-local for X=∅because F= LFX(f), where f(S) = Ffor each S∈J. (2) If X=J, the class of all simple groups, an X-formation function is simply a formation function and the X-local formations are exactly the local formations. (3) If X=P, the class of all abelian simple groups, then an X-formation function is a Baer function and the X-local formations are exactly the Baer-local ones. We will need the following results throughout the paper. Lemma 3. Let Fbe an X-local formation and let fbe an X-formation function defining F. (1) If Vis a normal subgroup of Usuch that V∈EX,U/V ∈F, and U/ CU(V)∈f(p)for every p∈π(V), then U∈F. (2) If fis integrated, then Spf(p)⊆Ffor every p∈char(X). 4 If Kis a class of groups and p∈char X, denote KX(p) := SpQ R0U/ CU(V/W)|U∈Kand V/W is an Xp-chief factor of U, taking into account that KX(p) = ∅if there does not exist any group U∈K with an Xp-chief factor. We write formX(K) to denote the smallest X-local formation containing K, i. e., the intersection of all X-local formations containing K. If X=J, we write lform(K) to denote formX(K). The following theorem describes a full and integrated X-formation function defining formX(K). Theorem 4 ([4] and [9]). (1) Let Kbe a class of groups. Then formX(K) = LFX(K), where Kis the following X-formation function:    K(p) = KX(p)if p∈char(X) K(E) = Q R0(K)if E∈X0 Moreover, Kis full and integrated. We say that Kis the canonical X-local definition of formX(K). (2) If Xand Xare two classes of simple groups such that X⊆Xand Fis an X-local formation, then Fis X-local. (3) If Fis an X-local formation defined by a full and integrated X-formation function f, then f(p) = (U|CpoU∈F)for every p∈char(X). In particular, if Kis a class of groups, then KX(p) = U|CpoU∈ formX(K)for every p∈char(X). Lemma 5. Consider p∈char X. If Kis a quotient-closed class of groups, then KX(p) := SpQ R0U/ CU(V)|Uis monolithic, U∈K, and V= Soc(U)∈EXp. PROOF. Consider a group L∈Kand let M/N be an Xp-chief factor of L. Take Rmaximal among the normal subgroups Tof Lsuch that M∩T=N. Then the quotient group U=L/R ∈Kis monolithic and its minimal normal subgroup is V=MR/R ∼ =M/M∩R=M/N (see [9, Lemma 1.30]). Moreover, R≤CL(M/N) and L/ CL(M/N)∼ =U/ CU(V). Notation which is not explained here is consistent with the one in [1]. 5 3 Products of X-local formations The following example shows that the formation product of two X-local formations is not in general an X-local formation. Example 6 ([6]). Consider F=D0(1, A5), the formation composed of all groups that are direct products of copies of A5together with the trivial group, and G=S2. It is clear that Fand Gare Baer formations, that is, X-local where X=P. Assume that H=F◦Gis a P-local formation. By Theorem 4, we have that H= LFP(H), where    H(p) = HP(p) if p∈P H(E) = Hif E∈P0 Since G⊆H, it follows that H(2) 6=∅. Consider U= SL(2,5). Then U/ Z(U)∈ Hand U/ CUZ(U)∈H(2). Applying Lemma 3, we have that U∈H. This is not true. Hence His not a Baer-local formation. Taking the above example into account, it is natural to study conditions on two non-empty X-local formations Fand Gto ensure that F◦Gis an X-local formation. In the following Fand Gare non-empty formations and H=F◦G. The next theorem provides an X-local definition of formX(H). Theorem 7. Assume that H=F◦G, where Fand Gare non-empty formations, and Fis an X-local formation. Then the smallest X-local formation formX(H)containing His X-locally defined by the X-formation function hgiven by h(p) =    FX(p)◦Gif Sp⊆F GX(p)if Sp6⊆ Fp∈char X h(S) = HS∈X0 PROOF. We know by Theorem 4 that formX(H) = LFX(H), where His the X-formation function defined by    H(p) = HX(p) if p∈char(X) H(S) = Hif S∈X0 If we prove that H(p) = h(p) for every p∈char(X), then the result is clear. 6 By Lemma 5, we have that H(p) := SpQ R0U/ CU(V)|Uis monolithic, U∈H, and V= Soc(U)∈EXp. Assume that Uis a monolithic group in H, where V= Soc(U)∈EXp, and consider A=UG. If A= 1, it follows that U∈G. If Sp⊆F, then h(p) = FX(p)◦Gand, therefore, U∈h(p). If Sp6⊆ F, then U/ CU(V)∈GX(p) = h(p). Now suppose that A6= 1. Since V≤A, applying [1, A, 4.13], it follows that V=V1× · · · × Vn, where Viis a minimal normal subgroup of A, 1 ≤i≤n. Since A∈F, it follows that A/ CA(Vi)∈FX(p), for all i∈ {1, . . . , n}, and p| |V|. Consequently U/ CU(V)G∼ =A/ CA(V)∈R0FX(p) = FX(p) and so U/ CU(V)∈FX(p)◦G=h(p) for all p| |N|. It follows that H(p) = HX(p)⊆ SpQ R0h(p) = h(p). Now we prove that h(p)⊆H(p) = HX(p). If Sp6⊆ F, then clearly h(p) = GX(p)⊆HX(p). Suppose that Sp⊆F, that is, h(p) = FX(p)◦G. Consider a group U∈FX(p)◦G. Then the wreath product CpoU∈Sp(FX(p)◦G)⊆ SpFX(p)◦G. By Theorem 4, we know that SpFX(p)⊆Fand, hence, CpoU∈ F◦G=H⊆formX(H). Then Theorem 4 shows that U∈HX(p). This proves that h(p)⊆H(p). The following definition was introduced in [6] for Baer-local formations. Definition 8. Consider H=F◦G, where Fand Gare non-empty formations. We say that the boundary b(H) of His XG-free if every group U∈b(H) such that Soc(U) is a p-group for some prime p∈char Xsatisfies that U/ CUSoc(U)/∈GX(p). Remark 9.Note that in Example 6, b(H) is not PG-free. Lemma 10. If Kis a formation and U∈b(K)∩formX(K), with V= Soc(U), then Vis an abelian p-group for a prime p∈char(X). PROOF. According to Theorem 4, formX(K) = LFX(K), where Kis the following X-formation function:    K(p) = KX(p) if p∈char(X) K(E) = Kif E∈X0 Clearly, Vis a minimal normal subgroup of U. If Vwere an X0-group, we would have that U∈K(E) for some E∈X0. This would imply that U∈ K, contrary to supposition. Hence Vis an X-chief factor of U. Let pbe a 7 In [16], it is shown that ω-saturated formations are Xω-local, where Xωis the class of all simple ω-groups. However, the converse does not hold. The following lemma gives a characterization of p-saturated formations. Lemma 26. Let Kbe a non-empty formation. Then Kis p-saturated if and only if KJ(p)⊆K. PROOF. Suppose that Kis a p-saturated formation, where pis a prime. Then lform(K)⊆Np0K. By Theorem 4, we know that KJ(p)⊆lform(K) and, therefore, KJ(p)⊆Np0K. This implies that KJ(p)⊆K. Now suppose that Kis not p-saturated and KJ(p)⊆K. Let Ube a group of minimal order satisfying U/Φ(U)∩Op(U)∈Kand U /∈K. Then U is a monolithic group and V:= Soc(U)≤Φ(U)∩Op(U). We have that Op0,p(U/V ) = Op0,p(U)/V , since V≤Φ(U). Moreover, U/V ∈Kand, therefore, U/ Op0,p(U)∈KJ(p), bearing in mind that p∈π(U/V ). Since Op0,p(U) = Op(U), U∈KJ(p)⊆K. This is not possible. Theorem 27. Suppose that H=F◦G, where Fand Gare non-empty formations, and let pbe a prime. Then the following statements are equivalent: (1) His a p-saturated formation. (2) Either HJ(p)⊆Gor SpHJ(p)G⊆F. PROOF. It is clear applying Lemma 26 and Lemma 20 for X=J. Theorem 27 also confirms a more general version of the abovementioned conjecture of Shemetkov concerning the non-decomposability of the formation of all p-groups (pa prime) as formation product of two non-trivial subformations. Corollary 28. Suppose that H=F◦G, where Fand Gare non-empty formations, and His p-saturated. If F⊆Spand F6=Sp, then H=G. PROOF. If HJ(p) = ∅, it follows that H⊆Ep0. In this case, we have that H⊆Ep0∩(Sp◦G). Therefore, H⊆G. If HJ(p)6=∅, we have that H⊆Ep0HJ(p). If HJ(p) is contained in G, then H⊆Ep0HJ(p)∩(SpG)⊆(Ep0G)∩(SpG) = Gand the result holds. Suppose that HJ(p) is not contained in G. Then SpHJ(p)Gis contained in Fby Theorem 27. In particular, Sp⊆F, and we have a contradiction. 14 Acknowledgements The authors are indebted to the referee for his/her helpful comments, which have improved the exposition of the paper. This research is supported by Grant MTM2004-08219-C02-02, from MEC (Spain) and FEDER (European Union). The second author has been supported by a predoctoral grant from MEC (Spain). References [1] K. Doerk, T. Hawkes, Finite Soluble Groups, no. 4 in De Gruyter Expositions in Mathematics, Walter de Gruyter, Berlin, New York, 1992. [2] L. A. Shemetkov, Radical and residual classes of finite groups, in: Y. Bahturin (Ed.), Algebra. Proceedings of the International Algebraic Conference on the occasion of the 90th birthday of A. G. Kurosh, Moscow, Russia, May 25–30, 1998, Walter de Gruyter, Berlin, New York, 2000, pp. 331–344. [3] P. F¨orster, Projective Klassen endlicher Gruppen IIa. Ges¨attigte Formationen: ein allgemeiner Satz von Gasch¨utz-Lubeseder-Baer-Typ, Publ. Sec. Mat. Univ. Aut`onoma Barcelona 29 (2–3) (1985) 39–76. [4] A. Ballester-Bolinches, C. Calvo, R. Esteban-Romero, On X-saturated formations of finite groups, Comm. Algebra 33 (2005) 1053–1064. [5] A. N. Skiba, L. A. Shemetkov, Multiply ω-local formations and Fitting classes of finite groups, Sib. Adv. Math. 10 (2) (2000) 112–141. [6] E. Salomon, ¨ Uber lokale und Baerlokale Formationen endlicher Gruppen, Master’s thesis, Johannes Gutenberg-Universit¨at Mainz, Mainz, Germany (1983). [7] V. D. Mazurov, E. I. Khukhro (Eds.), Unsolved Problems in Group Theory. The Kourovka Notebook, 11th Edition, Institute of Mathematics, Sov. Akad. Nauk SSSR, Siberian Branch, Novosibirsk, SSSR, 1990. [8] L. A. Shemetkov, A. N. Skiba, On hereditarily indecomposable formations of groups, Dokl. Akad. Nauk BSSR 33 (7) (1989) 581–582, russian. [9] C. Calvo L´opez, Formaciones X-locales de grupos finitos, Master’s thesis, Facultat de Matem`atiques, Universitat de Val`encia (2004). [10] M. Suzuki, Group theory I, Vol. 247 of Grundlehren Math. Wiss., SpringerVerlag, Berlin-Heidelberg-New York, 1982. 15 [11] A. Ballester-Bolinches, M. D. P´erez-Ramos, Some questions of the Kourovka Notebook concerning formation products, Comm. Algebra 26 (5) (1998) 1581– 1587. [12] V. A. Vedernikov, On some classes of finite groups, Dokl. Acad. Nauk BSSR 32 (10) (1988) 872–875. [13] N. T. Vorob’ev, On factorizations of non-local formations of finite groups, Vopr. Algebry 6 (1993) 21–24. [14] B. Huppert, Endliche Gruppen I, Vol. 134 of Grundlehren Math. Wiss., Springer-Verlag, Berlin, Heidelberg, New York, 1967. [15] L. A. Shemetkov, On the product of formations, Dokl. Akad. Nauk BSSR 28 (2) (1984) 101–103. [16] A. Ballester-Bolinches, C. Calvo, R. Esteban-Romero, A question of the Kourovka Notebook on formation products, Bull. Austral. Math. Soc. 68 (2003) 461–470. 16