On X-saturated formations of finite groups

Document downloaded from: This paper must be cited as: The final publication is available at Copyright Additional Information Taylor and Francis Ballester Bolinches, A.; Calvo Lopez, C.; Esteban Romero, R. (2005). On X-saturated formations of finite groups. Communications in Algebra. 4(33). doi:10.1081/AGB-200053809 http://www.tandfonline.com/10.1081/AGB-200053809 http://hdl.handle.net/10251/19032 This is an Author's Original Manuscript of an article submitted for consideration in Communications in Algebra 33(4):1053-1064 (2005) (copyright Taylor & Francis). Communications in Algebra is available online at http://www.tandfonline.com/10.1081/AGB-200053809 DOI 10.1081/AGB-200053809 The final publication is available at link.springer.com This is an Author’s Original Manuscript of an article submitted for consideration in Communications in Algebra 33(4):1053–1064 (2005) (copyright Taylor & Francis). Communications in Algebra is available online at http://www.tandfonline.com/10.1081/AGB-200053809 DOI 10.1081/AGB-200053809 ON X-SATURATED FORMATIONS OF FINITE GROUPS A. Ballester-Bolinches1, Clara Calvo1, and R. Esteban-Romero2 1Departament d’Àlgebra, Universitat de València, Dr. Moliner, 50; E-46100 Burjassot (València), Spain email: [email protected] and [email protected] 2Departament de Matemàtica Aplicada, Universitat Politècnica de València; Camí de Vera, s/n, E-46022 València, Spain email: [email protected].es Mathematics Subject Classfication (2000): 20D10, 20F17 ABSTRACT In the paper, a Frattini-like subgroup associated with a class Xof simple groups is introduced and analysed. The corresponding X-saturated formations are exactly the X-local ones introduced by Förster. Our techniques are also very useful to highlight the properties and behaviour of ω-local formations. In fact, extensions and improvements of several results of Shemetkov are natural consequences of our study. 1 INTRODUCTION In the sequel it is understood that all groups are finite. Recall that a formation is a class of groups which is closed under taking epimorphic images and subdirect products. A formation Fis said to be saturated if 1 the condition G/Φ(G)∈Falways implies G∈F. Saturated formations play a very important role in the theory of classes of groups because they are exactly those formations Ffor which every group has F-projectors. The family of local formations introduced by Gaschütz allows us to generate saturated formations of ever-increasing complexity. In fact, both families coincide. It was proved by Gaschütz and Lubeseder in the soluble universe, and by Schmid in the general one. Today these results are known as Gaschütz-Lubeseder-Schmid theorem (see [1; Section IV] for details). In an unpublished manuscript, Baer has studied a different definition of local formation. It is more flexible than the original one because the simple components, rather than the primes dividing its order, are used to label a chief factor and its automorphism group. Therefore the actions on the insoluble chief factors can be independent of those on the abelian chief factors (see [1; Section IV] for details). 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. The presentation of Baer’s theorem in [1; IV,4.17] includes a nice variation of the proof of Lubeseder’s theorem in the soluble universe. Baer-local formations were also considered by Shemetkov (see [2]). He calls them composition formations. With the purpose of presenting a common extension of the Gaschütz-Lubeseder-Schmid and Baer theorems, Förster introduced the concept of X-local formation, where Xis a class of simple groups with a completeness property ([3]). If X=J, the class of all simple groups, X-local formations are exactly the local formations. When X=P, the class of all abelian simple groups, the notion of X-local formation coincides with the concept of Baer-local formation. Förster also defined a Frattini-like subgroup Φ∗ X(G)for each group G, which enables him to introduce the concept of X-saturation. He proves that X-saturated formations are exactly the X-local ones. However, Förster’s definition of X-saturation is not the natural one if our aim is to generalise the concepts of saturation and soluble saturation. Since OJ(G) = Gand OP(G) = GS, we would expect the X-Frattini subgroup of a group Gto be defined as ΦOX(G), where OX(G)is the largest normal subgroup of Gwhose composition factors belong to X. In many cases Φ∗ X(G)does not coincide with ΦOX(G). Since ΦOX(G)≤Φ∗ X(G)for every group G, we can deduce from Förster’s theorem that every X-local formation F fulfills the following property: A group Gbelongs to Fif and only if G/ΦOX(G)belongs to F. (*) 2 Therefore from the very beginning the following question naturally arises: Open question 1.1. Let Fbe a formation with the property (*). Is F X-local? One of the purposes of the present paper is to draw near the solution of the above question. We introduce another X-Frattini subgroup which is smaller than Förster’s one. We will prove that X-local formations are exactly the X-saturated ones defined using our X-Frattini subgroup. The method used in the proof of this theorem can be applied to deduce that ω-saturated formations are ω-local. We will also get the main results of [4] as particular cases of ours. For the basic definitions and results on the theory of formations, as well as for the standard notations, the reader is referred to the book of Doerk and Hawkes [1]. 2 PRELIMINARIES We begin with the concept of X-local formation due to Förster [3]. Denote by Jthe class of all simple groups. For any subclass Yof J, we put Y0=J\Y. Denote by EYthe class of groups whose composition factors belong to Y. It is clear that EYis a Fitting class, and so each group Ghas a largest normal EY-subgroup, the EY-radical OY(G). A chief factor of Gwhich belongs to EY is called a Y-chief factor, and if, moreover, pdivides the order of a Y-chief factor H/K of G, we shall say that H/K is a Yp-chief factor. In the sequel it will be convenient to identify the prime pwith the cyclic group Cpof order p. Throughout this paper, Xdenotes a fixed class of simple groups satisfying π(X) := {p∈P|there exists G∈Xsuch that pdivides |G|} ={p∈P|Cp∈X}=: char X. Definition 2.1 (Förster).An X-formation function fassociates to each X∈ char(X)∪X0a formation f(X)(possibly empty). If fis an X-formation function, then the X-local formation LFX(f)defined by fis the class of all groups G satisfying the following two conditions: 1. if H/K is an Xp-chief factor of G, then G/ CG(H/K)∈f(p), and 2. if G/K is a monolithic quotient of Gsuch that the composition factor of its socle Soc(G/K)is isomorphic to E∈X0, then G/K ∈f(E). 3 A formation Fis said to be X-local if there exists an X-formation function fsuch that F= LFX(f). Förster also defined an X-Frattini subgroup Φ∗ X(G)of a group G: Definition 2.2 (Förster).Let Gbe a group. •For a prime p, we define Φp X(G): –If Op0(G) = 1, Φp X(G) := (Φ(G)if SocG/Φ(G)and Φ(G)belong to EX, ΦOX(G)otherwise. –In general, Φp X(G)is the subgroup of Gsuch that Φp X(G)/Op0(G)=Φp XG/ Op0(G). •Finally Φ∗ X(G) := OX(G)∩ \ p∈char(X) Φp X(G) . As it is announced, we introduce another X-Frattini subgroup ΦX(G)of a group G. Definition 2.3. 1. Let pbe a prime number. We say that a group Gbelongs to the class AXp(P2)provided there exists an elementary abelian normal p-subgroup Nof Gsuch that (a) N≤Φ(G)and G/N is a primitive group with a unique non-abelian minimal normal subgroup, i. e., G/N is a primitive group of type 2, (b) Soc(G/N)∈EX\Ep0, and (c) Ch G(N)≤N, where Ch G(N) := \{CG(H/K)|H/K is a chief factor of Gbelow N}. 2. The X-Frattini subgroup of a group Gis the subgroup ΦX(G)defined as ΦX(G) := (ΦOX(G)if G /∈AXp(P2)for all p∈char X, Φ(G)otherwise. 4 Note that in the case X=J, the class of all simple groups, ΦX(G) = Φ(G) for every group G, and in the case X=P, the class of all abelian simple groups, ΦX(G) = Φ(GS)for every group G, where GSis the soluble radical of G. It can be proved that ΦX(G)≤Φ∗ X(G)for every group G, but the equality is not true in general. In fact, for every class of simple groups X6=J, there exists a group Gsuch that ΦX(G)<Φ∗ X(G), as it is shown in the following example. Example 2.4. If X6=J, it follows that there exists a non-abelian simple group E and a prime p∈π(E)such that E∈X0and p∈char(X). Let Tbe the group algebra GF(p)E. Consider G= [T]E, the corresponding semidirect product. It is rather clear that Φ(G) = Rad(T). We have that Φp P(G) = Φ(G), since Op0(G)=1and the groups Φ(G)and SocG/Φ(G)belong to EX. If q6=p, then Φ(G)≤T≤Oq0(G)≤Φq P(G). Now it follows that Φ∗ P(G) = Φ(G). Since p∈π(E), we have by Maschke’s theorem that Rad(T)6= 1. Therefore Φ∗ X(G)6= 1. Clearly, G /∈AXq(P2)for all q∈char(X). Hence, ΦX(G) = ΦOX(G)= Φ(T)=1. Definition 2.5. A formation Fis said to be X-saturated whenever the condition G/ΦX(G)∈Falways implies G∈F. There exist groups Gfor which ΦOX(G)is a proper subgroup of ΦX(G), as the following example, suggested by John Cossey, shows: Example 2.6. Let Xbe the maximal Frattini extension of the alternating group A5 of degree 5corresponding to the prime p= 5. Then Xhas an elementary abelian normal subgroup Mof order 53contained in Φ(X)and X/M is isomorphic to A5 (see [1; Appendix β]). Let Tbe a non-abelian group of order 55, and consider Y=XoT. Let Gbe a subdirect product of Yand a cyclic group of order 25 with amalgamated factor group isomorphic to C5. Consider the class X= (A5, C2, C3, C5). Then Gbelongs to AX5(P2), and so ΦX(G) = Φ(G), which has order (53)55 ·5. On the other hand, ΦOX(G)has order (53)55. 3 THE X-SATURATED FORMATIONS ARE X-LOCAL The main goal in this section is to prove: Theorem A. The X-local formations are exactly the X-saturated formations. The following series of lemmas on X-saturated formations is needed to prove Theorem A. 5 Lemma 3.1. Let Fbe an X-saturated formation, Xa group, and pa prime in char(X). If there exists a faithful X-module Mover GF(p)such that [M]X∈F, then [N]X∈Ffor every irreducible GF(p)X-module N. Proof. We can argue as in [1; IV,4.1], bearing in mind that the Hartley group used in the proof is a p-group and hence it belongs to EX. Lemma 3.2. Let Fbe an X-saturated formation, Ga group and let pbe a prime in char X. If Cp∈Fand Nis a normal elementary abelian p-subgroup of Gsuch that and [N](G/N)∈F, then G∈F. Proof. Analogous to [1; IV,4.15], noting that the Hartley group is a p-group as in the previous lemma. Lemma 3.3. Let Fbe an X-saturated formation and pa prime in char X. If X∈ R0G/ CG(H/K)|G∈Fand H/K is an Xp-chief factor of G, then [N]X∈F for every irreducible GF(p)X-module N. Proof. By Lemma 3.1, it is enough to find a faithful X-module Mover GF(p) such that [M]X∈F. Since X∈R0G/ CG(H/K)|G∈Fand H/K is an Xp-chief factor of G, there exist a natural number nand normal subgroups Xiof X, for i= 1,2, . . . , n, such that Tn i=1 Xi= 1 and X/Xi∼ =Gi/CGi(Hi/Ki), where Gi∈Fand Hi/Ki is an Xp-chief factor of Gi. Assume that Hi/Kiis non-abelian for some i= 1,2, . . . , n. Then G:= Gi/CGi(Hi/Ki)is a primitive group of type 2. Consider the maximal Frattini extension Eof Gcorresponding to the prime p. Then Ehas a elementary abelian normal p-subgroup Ap(G), the Frattini p-module of G, such that E/ Ap(G)∼ =G.Ap(G)can be regarded as a GF(p)G-module and so viewed we have that KerGon SocAp(G)= Op0,p(G)(cf. [1; Appendix β]). In this case Op0,p(G) = 1 and, therefore, there exists an irreducible GF(p)G-submodule of Ap(G), say T, such that CG(T)=1. Note that E∈AXp(P2). This means that ΦX(E) = Φ(E)=Ap(G)and, therefore, E/ΦX(E)∼ =G∈F. Since Fis X-saturated, it follows that E∈F. We have that Tis an abelian Xp-chief factor of Esuch that E/ CE(T)∼ =Gand E∈F. Hence we can assume that Hi/Kiis abelian for all i. By [1; IV,1.5], it follows that [Hi/Ki]Gi/CGi(Hi/Ki)∈F. Consequently, [Hi/Ki](X/Xi)∈F. Consider now W:= [Hi/Ki]X. We have that W(Hi/Ki)∈Fand W/Xi∈ F. Therefore W∈R0F=F. 6 Write M:= H1/K1×H2/K2×· · ·×Hn/Kn. We have that Hi/Kiis a faithful Gi/CGi(Hi/Ki)-module over GF(p). Since Gi/CGi(Hi/Ki)∼ =X/Xi, we obtain that Hi/Kiis a GF(p)X-module and CX(Hi/Ki) = Xi. Since Tn i=1 Xi= 1, it follows that Mis a faithful X-module over GF(p). Moreover, [M]X∈R0F= F, as desired. Theorem 3.4. If Fis an X-saturated formation, then Fis X-local. Proof. Let fbe the X-formation function defined as f(X) =      Q R0G/ CG(H/K)|G∈Fand H/K is an Xp-chief factor of G if X=p∈char(X), Fif X∈X0. It is clear that F⊆LFX(f). Suppose that F6= LFX(f)and take a minimal group in LFX(f)\F. Clearly, Gis a monolithic group and N:= Soc(G)is an abelian X-chief factor of G. Let pbe the prime dividing the order of N. Thus, f(p)6=∅ and we can take a group Y∈R0G/ CG(H/K)|G∈Fand H/K is an Xp-chief factor of G. Consider the trivial GF(p)Y-module V. By Lemma 3.3, we have that [V]Y∼ = V×Y∈Fand thus Cp∼ =V∈F. Now we distinguish two cases: •If CG(N) = N, we have that G/N ∈f(p). Therefore there exist X∈ R0G/ CG(H/K)|G∈Fand H/K is an Xp-chief factor of Gand TE Xsuch that G/N ∼ =X/T. Ncan be regarded as an irreducible GF(p)G-module and as an irreducible GF(p)X-module. By Lemma 3.3, we have that [N]X∈F. This means that [N](X/T)and [N](G/N)also belong to F. We can now apply Lemma 3.2 to obtain that G∈F, a contradicction. •Assume now that N < CG(N). By [1; IV,1.5], the group B:= [N](G/N) belongs to LFX(f). Consider M:= CB(N)∩(G/N). Since M6= 1, the minimality of G implies that B/M ∈F. Moreover, since B/N ∈Fand M∩N= 1, we have that B∈R0F=F. By Lemma 3.2, we obtain that G∈F, a contradiction. We have proved that LFX(f)⊆Fand, thus, F= LFX(f). 7 Proof. Assume that the result is false and consider a group Gof least order such that G/ Oω(G)∩Φ(G)∈bform(F)and G /∈bform(F). We have that Gis a monolithic group whose socle is an ω-group. We know that Fis Xω-saturated by [6; Section 3], where Xωis the class of all simple ω-groups. By Corollary 4.10, Nω0Fis a Baer-local formation. Therefore G/ Oω(G)∩Φ(G)∈Nω0F, since bform(F)⊆Nω0F. By Lemma 4.12, it follows that Nω0Fis ω-saturated and, hence, G∈Nω0F. Bearing in mind that Oω0(G) = 1, we conclude G∈bform(F), a contradiction. As a particular case of Corollary 4.13, taking ω={p}, we get a result proved by Shemetkov [4; Theorem 3.1]. ACKNOWLEDGMENT This work has been supported by Proyecto BFM2001-1667-C03-03, MCyT (Spain) and FEDER (European Union). REFERENCES [1] Doerk, K.;Hawkes, T. Finite Soluble Groups. Number 4 in De Gruyter Expositions in Mathematics. Walter de Gruyter: Berlin, New York, 1992. [2] L. A. Shemetkov. Two directions in the development of the theory of nonsimple finite groups. 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