Nilpotent-like Fitting formations of finite soluble groups

Document downloaded from: This paper must be cited as: The final publication is available at Copyright Additional Information http://dx.doi.org/10.1017/S0004972700018943 http://hdl.handle.net/10251/43727 Cambridge University Press Ballester-Bolinches, A.; Pérez-Ramos, M.; Martínez Pastor, A. (2000). Nilpotent-like Fitting formations of finite soluble groups. Bulletin of the Australian Mathematical Society. 62(3):427-433. doi:10.1017/S0004972700018943 BULL. AUSTRAL. MATH. SOC. 20D10, 20F17 VOL. 62 (2000) [427-433] NILPOTENT-LIKE FITTING FORMATIONS OF FINITE SOLUBLE GROUPS A. BALLESTER-BOLINCHES, M.D. PEREZ-RAMOS AND A. MARTINEZ-PASTOR Dedicated to Professor K. Doerk on his 60th Birthday. In this paper the subnormal subgroup closed saturated formations of finite soluble groups containing nilpotent groups are fully characterised by means of extensions of well-known properties enjoyed by the formation of all nilpotent groups. 1. INTRODUCTION AND NOTATION All groups considered in the paper will be finite and soluble. The class of groups whose members are a direct product of Hall subgroups corresponding to a given partition of the set of all primes is a subgroup-closed saturated formation enjoying many interesting properties of nilpotent type. In fact, this family of formations was characterised in [2] as the subgroup-closed saturated formations T for which the set of all ^"-subnormal subgroups of each group G is a sublattice of the subgroup lattice of G. Hence the members of this family are called lattice formations. The purpose of this paper is to prove that the majority of properties of nilpotent type enjoyed by the lattice formations actually characterise them (Theorem 1). On the other hand, it is known that the Fitting subgroup of a group which is the product of two nilpotent subgroups inherits the factorisation [1, Lemma 2.5.7]. The natural question is now for which subgroup-closed Fitting formations T the ^-radical of a group G which is the product of two ^-groups inherits the factorisation. As application of Theorem 1, we see that this property also characterises the aforesaid subgroup-closed Fitting formations. This is Theorem 2 of the paper. We shall adhere to the notation and terminology employed in [5]. This book is our main reference for results concerning formations and Fitting classes. For the sake of completeness, we gather some notation and recall some definitions and results. We Received 24th February, 2000 This research has been supported by Proyecto PB 97-0674-C02-02 of DGICYT, Ministerio de Educaci6n y Ciencia, Spain. The authors thank J. Cossey for many helpful conversations during his visit to Valencia (Spain) supported by the Universidad Politecnica de Valencia. Copyright Clearance Centre, Inc. Serial-fee code: 0004-9727/00 SA2.00+0.00. 427 of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700018943 Downloaded from https://www.cambridge.org/core. Universitat Politecnica de Valencia, on 30 May 2018 at 11:19:18, subject to the Cambridge Core terms 428 A. Ballester-Bolinches, M.D. Perez-Ramos and A. Martinez-Pastor [2] denote by IP the set of all prime numbers and, for a prime p, Cp is the cyclic group of order p. For a group G, ir(G) denotes the set of all prime divisors of \G\. If X is a class of groups, the characteristic of X is charA" = {p € P | Cp e X}. The boundary b(X) of a class of groups X consists of all groups G satisfying G £ X and G/N € X for all 1 ^ TV <3 G. If 7r is a set of prime numbers, let 5 and S* denote the classes of soluble and soluble 7r-groups, respectively. The well-known Gaschutz-Lubeseder-Schmid Theorem states that in the general finite universe, saturated formations are exactly local formations, that is, formations F - LF(f) defined by a formation function /: LF{f) = (G e S | if H/K is a chief factor of G and p € TT(H/K), then G/CG(H/K) e f(p)), where £ is the class of all finite groups. In this case, / is said to be a local definition of T. Among all possible local definitions of a local formation T there exists exactly one, denoted by F, such that F is integrated (that is, F(p) C T for all p 6 P) and full (that is, SpF(p) = F(p) for all p e P). F is called the canonical local definition of T. If T is a Fitting class, a subgroup V of a group G is called an .F-injector of G if V n TV is an ^"-maximal subgroup of TV for every subnormal subgroup TV of G. By a well-known result due to Fischer, Gaschiitz and Hartley [5, IX, Theorem 1.4], every soluble group G has a unique conjugacy class of .F-injectors. It is also known that a subgroup-closed Fitting class of finite soluble groups is a saturated formation (Bryce and Cossey, [5, XI, Theorem 1.1]). 2. THE RESULTS LEMMA 1. Let T be a subgroup-closed Fitting class containing Af, the class of nilpotent groups. Then the following statements are equivalent: (i) Let F be the full and integrated local formation function defining T. Then there exists a partition {7r;} ie/ off such that F(p) — Sni for every prime number p € 7Tj and for every i & I. (ii) For each prime number peP, every primitive group G € T D b(F(p)) is cyclic. PROOF: (i) implies (ii). By applying [2, Lemma 3.2], a group G is an .F-group if and only G has a normal Hall ^-subgroup, for every i e I. Therefore, it is clear that (ii) holds. (ii) implies (i). Arguing as in (2) and (3) of [2, Theorem 3.3], we obtain that if p and q are two prime numbers such that p € charF(g), then charF(p) = charF(g). Now, by [5, IV,Lemma 3.16], F(p) is a subgroup-closed Fitting class for every prime p. Hence, if p,g G P and p€ charF(g), then 5P C F{q). We see that if p € P and n{p) - charF(p), then F(p) = 5 ff(p) D T. Since F(p) is a subgroup-closed Fitting class and F is integrated, it follows that F(p) C S,,^) n T, of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700018943 Downloaded from https://www.cambridge.org/core. Universitat Politecnica de Valencia, on 30 May 2018 at 11:19:18, subject to the Cambridge Core terms [3] Finite soluble groups 429 for every prime p. Assume that there exists a prime p such that F(p) / Sv(p) n T and consider a group G € (£*(?) n T)\F(p) of minimal order. We have that G is a primitive group, because F(p) is a saturated formation. Since G G Tr\b(F(p)), then G is cyclic by (ii). Therefore G is a g-group with q € n(p) and so G € F(p) because F(p) is a Fitting class, a contradiction. Consequently, if p and q are two prime numbers such that p € charF(g), then F(p) = F(g). Finally, we claim that for every prime p if 7r(p) = charF(p), then F(p) = <SV(P). It is clear that F(p) C 5^(p). If F(p) 7^ ^(p), we choose a group G of minimal order in <Sir(p)\F(p). Then G is again a primitive group, Soc(G) is a g-group, with q € n(p), and G/Soc{G) e F(p). Consequently, G £ SqF(p) = 5,F(g) = F(g) = F(p), a contradiction. D Recall that if T is a saturated formation and G is a group, a maximal subgroup M of G is said to be .F-normal in G if G/CoiedM) £ T. A subgroup H of G is called ^-subnormal in G if either H = G or there exists a chain H = Ho ^ H\ ^ • • • ^ //„ = G such that Hi is an J"-normal maximal subgroup of Hi+i for 0 ^ i < n. It is clear that if T = A/", the saturated formation of nilpotent groups, the ^"-subnormal subgroups of G are exactly the subnormal subgroups of G. THEOREM 1. Let T be an sn-closed saturated formation containing A/". Then the following statements are pairwise equivalent: (1) Let F be the full and integrated local formation function defining T. Then there exists a partition {fti}i£i of P such that F(p) = S^ for every prime number p 6 TTJ and for every i £ I. (2) If H and K are two T-subnormal subgroups of a group G, then (H, K)r = (3) T is a Fitting class satisfying that ifG is a group, V is an T-injector ofG and H is an 7-subnormal subgroup of G, then V C\H is an 7-injector of H. (4) T is a Fitting class such that the T-radical G? of a group G is: Gr = (X€T\Xis T-subnormal in G). (5) IfH and K are two T-subnormal T-subgroups of a group G, then (H, K) e T. (6) T is a Fitting class and if H is an T-subnormal T-subgroup of a group G, then (H, H9) 6 T for every g€G. PROOF: First we claim that if T is a saturated Fitting formation containing M and satisfying (6), then T is subgroup-closed. Denote Ts = (X \ S{X) C T), where S(X) = (H € 5 | H ^ X). We prove that T = TsSuppose not and let G be a group of minimal order in T \ Ts. Then there of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700018943 Downloaded from https://www.cambridge.org/core. Universitat Politecnica de Valencia, on 30 May 2018 at 11:19:18, subject to the Cambridge Core terms 430 A. Ballester-Bolinches, M.D. Perez-Ramos and A. Martinez-Pastor [4] exists a subgroup H of G such that H £ T. We choose H of minimal order. Then every proper subgroup of H is in T. By the minimal choice of G, it is clear that HK/K 6 F for every proper normal subgroup K of G. In particular, G has a unique minimal normal subgroup, N say. Let p be the prime dividing \N\. Since H/H C\N £ F, it is clear that H7 is a p-group. We claim that H/H7 has a unique maximal subgroup, and so H/H7 is a cyclic <7-group for some prime q ^ p. First we shall show that H/H7 is a nilpotent group. Suppose not and let M/H7 be a maximal subgroup of H/H7 such that M/H7 is not a normal subgroup. Then there exists g € H such that M/H7 ^ M9/H7. Notice that M £f by the choice of #. Moreover, M is an F-subnormal subgroup of H since H7 ^ M. By applying (6), we have that H = (M, M9) 6 F, a contradiction. Hence any maximal subgroup of H/H7 is a normal subgroup and so H/H7 e Af. Assume now that there exist two maximal subgroups of H/H7, M\/H7 and M2/H7, such that Mi ^ M2. Then Mj is a normal subgroup of H for i = 1,2 and M* € F for i = 1,2 by the choice of H. Since F is a Fitting class it follows that H = M X M 2 € F, a contradiction. Therefore we have /f = H q H 7 where Hq is a Sylow g-subgroup of H. Notice that G/C G {N) 6 F(p) since G € .F. If #, < CG(7V), we have H = Hq x H7 G N Q T, a contradiction. Hence g £ n(G/Ca(N)). This is to say that q s charF(p) and so Sq Q F{p) because F(p) is a Fitting class by [5, IV,Lemma 3.16]. Therefore H = H 7 H q € SpF(p) = F{p) C T since F is full and integrated, a contradiction. Notice now that if T is a saturated Fitting formation satisfying either (3) or (4), we can apply similar arguments to those used above to conclude that T is also subgroupclosed. (1) equivalent to (2). First observe that if T is an sn-closed saturated formation satisfying (2), then it is a Fitting class since every subnormal subgroup is ^-subnormal. Hence T satisfies (6) and it is subgroup-closed because of the previous proof. Consequently, (1) equivalent to (2) is [4, Theorem 2]. (1) implies (3). This is [2, Theorem 4.5]. (3) implies (4). Let X be an .F-subnormal ^-subgroup of G. If V is an .F-injector of G, then VnX = X by (3). Hence X is contained in every F-injector V of G and then X ^ fi^^j. geG It is clear that (4) implies (5) and (5) implies (6). (6) implies (1). By Lemma 1 it is enough to show that every primitive group G e Tnb(F{p)) is cyclic. It is clear that G has a unique minimal normal subgroup N, and N is a g-group, where q is a prime number, q / p. Then there exists an irreducible and faithful G-module Vp over GF(p). We claim that G has a unique maximal subgroup M such that CoreG(M) = 1. Assume that Mi and M2 are maximal subgroups of G, Mj ^ M2 and Corec(Mj) = 1, of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700018943 Downloaded from https://www.cambridge.org/core. Universitat Politecnica de Valencia, on 30 May 2018 at 11:19:18, subject to the Cambridge Core terms [5] Finite soluble groups 431 i = 1,2. Then Mj G F(p). Consider now the semidirect product H = [VP]G with respect to the action of G on Vp. Clearly H & T, so H* - Vp. On the other hand, for i = 1,2, VpMi is an ^"-normal maximal subgroup of H and V p M t G SpF(p) = F(p) C T. Now H = {VPMX,VPM2) = (VPMU [VpMi)') for some g G H. By applying (6), we have ff«f, a contradiction. Consequently, G is cyclic and the circle of implications is complete. D We say that a saturated Fitting formation containing TV is a lattice-formation if it satisfies one of the conditions of Theorem 1. 3. SOME APPLICATIONS In this section we give some applications of our main result in the framework of factorised groups. We recall here some definitions and elementary facts about these groups (see [1]). If a group G is the product of two subgroups A and B, we say that a subgroup S is factorised \i S = (AnS)(BnS) and AnB ^ S. If N is a normal subgroup of a factorised group G = AB, then there exists a subgroup X(N), called the factoriser of N in G = AB, which is the smallest factorised subgroup of G containing N and it has the form X{N) = ANnBN = {An BN)N = {Bn AN)N = (An BN){B n AN). This fact motivates the study of trifactorised groups G = AB — AK = BK, where K is a normal subgroup of G. It is well-known that a triply factorised group G = AB = AK = BK of two subgroups A and B and a normal subgroup K is nilpotent provided that A, B and K are nilpotent (see [1, Corollary 1.3.5]). The next result shows that the same is true for lattice-formations. PROPOSITION 1. Let T be a lattice-formation. Let G be a group of the form G = AB — AK = BK, where A and B are subgroups ofG and K is a normal subgroup ofG. If A, B and K are T-groups, then G is also an T-group. PROOF: First notice that, although G is assumed soluble, the solubility of G follows automatically from the conditions on the subgroups A, B and K. Let {TTi}i£i be a partition of the set P of all primes and let T be the formation locally defined by the integrated and full formation function F given by F(p) = Sni for every prime number p G TTJ and for every i £ I. Assume that the result is false and choose for G a counterexample of smallest order. It is clear that G is in the boundary of T. Consequently G must be a primitive group with a unique minimal normal subgroup TV. Let p be the prime dividing the order of iV and let i € I such that p G TTJ. NOW, since K is a normal .F-subgroup of G, if j G / and j ^ i, we have that Onj(K) ^ CG(N) ^ N. So Onj{K) = 1 and K is a 7rrgroup. Let A^ and B^ be the Hall 7r,'-subgroups of A and B. Since G ^ T, it follows that G is not a TTj-group. So we can assume that Aj. ^ 1. On the other hand it is clear that of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700018943 Downloaded from https://www.cambridge.org/core. Universitat Politecnica de Valencia, on 30 May 2018 at 11:19:18, subject to the Cambridge Core terms 432 A. Ballester-Bolinches, M.D. Perez-Ramos and A. Martinez-Pastor [6] A^.Bl, = B°,Aj. for every element g € G. So [A%B%] = 1 by [1, Lemma 2.5.1]. Since A°, ± 1, we "have that N ^ A% and so Bj. < C'G(N) < N. This implies that B is a TTj-group and so is G, a contradiction. D The previous result is a particular case of a more general one which will appear in [3]. THEOREM 2. Let T be a subgroup-closed Fitting formation containingM. Then the following statements are equivalent: (1) T is a lattice-formation. (2) T satisfies the property: (•) IfG = AB and A and B are T-groups, then the T-radical G? of G is a factorised subgroup, that is, G? = (GpC\A){G?C\B) and ACiB is contained in Gjr. PROOF: (1) implies (2). Let {7i"i};e/ be a partition of the set P of all primes and assume that T is the saturated Fitting formation locally denned by the integrated and full formation function F given by F(p) = 5W|. for every prime number p € 7TJ and for every is/. By induction on the order of G, we may suppose that the ^"-radical L/Gjr of the factor group GjGT = (AG?/Gj:)(BGr/Gjr) is factorised. Therefore GT < X(GF) = AGf D BGT ^ L and hence X(Gjr) is an .F-subnormal subgroup of G since X(Gjr)/Gr is ^-subnormal in L/GP e T. Moreover X{GT) = (An BGjr)GF = (B <1 AGT)GT = (A n BGjr)(B n AG?) is an .F-group by Proposition 1. Hence by Theorem 1, we have that X(Gjr) = G^ and Gjis factorised. (2) implies (1). We prove that every primitive group G 6 TTl b(F(p)) is cyclic and then we apply Lemma 1 to obtain the result. Let N be the unique minimal normal subgroup of the primitive group G € .Fn&(F(p)). Then TV is a g-group, where q is a prime number, q ^ p. If we assume that G is not a cyclic group, there exists M ^ 1, M € F(p) such that G is the semidirect product G — [N)M. There exists an irreducible and faithful G-module Vp over GF(p). Consider now the semidirect product H = [VP]G with respect to the action of G on Vp. Then H = (VPM)(NM) where VPM € SpF(p) = F(p) C T and G = NM € T. By (2), KT is a factorised subgroup, so M = VPM n NM ^ Hr. On the other hand, since H? D G is a non-trivial normal subgroup of G, we have N ^ Hjr nG^ //^r. Thus G = [N]M ^ HT. But then 7/ = [VP]G efandGe ^(p), a contradiction. D REFERENCES [1] B. Amberg, S. Pranciosi and F. de Giovanni, Products of Groups (Clarendon Press, Oxford, 1992). [2] A. Ballester-Bolinches, K. Doerk and M.D. Perez-Ramos, 'On the lattice of ^"-subnormal subgroups', J. Algebra 148 (1992), 42-52. of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700018943 Downloaded from https://www.cambridge.org/core. Universitat Politecnica de Valencia, on 30 May 2018 at 11:19:18, subject to the Cambridge Core terms [7] Finite soluble groups 433 [3] A. Ballester-Bolinches, A. Martinez-Pastor and M.C. Pedraza-Aguilera, 'Finite trifactorized groups and formations', J. Algebra 226 (2000), 990-1000. [4] A. Ballester-Bolinches, M.C. Pedraza-Aguilera and M.D. Perez-Ramos, 'On ^"-subnormal subgroups and ^-residuals of finite soluble groups', J. Algebra 186 (1996), 314-322. [5] K. Doerk and T. Hawkes, Finite soluble groups (Walter De Gruyter, Berlin, New York, 1992). Departament d'Algebra Departament d'Algebra Universitat de Valencia Universitat de Valencia C/ Doctor Moliner 50 C/ Doctor Moliner 50 46100 Burjassot (Valencia) 46100 Burjassot (Valencia) Spain Spain Escuela Universitaria de Informatica Departamento de Matematica Aplicada Universidad Politecnica de Valencia Camino de Vera, s/n 46071 Valencia Spain of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700018943 Downloaded from https://www.cambridge.org/core. Universitat Politecnica de Valencia, on 30 May 2018 at 11:19:18, subject to the Cambridge Core terms