On Lockett pairs and Lockett conjecture for π-soluble Fitting classes

Transcription

1 On Lockett pars and Lockett conjecture for π-soluble Fttng classes Lujn Zhu Department of Mathematcs, Yangzhou Unversty, Yangzhou , P.R. Chna E-mal: Nanyng Yang School of Mathematcs and Scence, Unversty of Scence and Technology of Chna, Hefe , P. R. Chna E-mal: N.T.Vorob ev Department of Mathematcs, Masherov Vtebsk State Unversty, Vtebsk , Belarus E-mal: Abstract In ths paper, we construct a new and wde famly of Lockett pars n the class of all fnte π- soluble groups and gve a new characterstc of the valdty of Lockett conjecture. As applcaton, some known results are followed. 1 Introducton In the theory of fnte soluble groups, many well known results related to research of structures of Fttng classes and canoncal subgroups are closed connected wth the operators and defned by Lockett [13] (see also [8, chapter X]). In fact, every nonempty Fttng class F has the assocated Fttng classes F and F, where F s the smallest Fttng class contanng F such that the F -radcal of the drect product G H of any two groups G and H s equal to the drect product of the F -radcal of G and the F -radcal of H; F s the ntersecton of all Fttng classes X such that X =F (see [13] or [8, Chapter X]). If F = F, then the Fttng class s called a Lockett class. The nterest for the research of F and F s determned manly by the followng crcumstances. Frstly, the famly of Fttng classes satsfyng F = F s vast. In fact, by [8, Theorem X.1.25], every Fttng class closed about homomorphc mages or closed about fnte subdrect products, and every Fscher class (see [8, IX.3.3]) are all Lockett classes. Secondly, Lockett [13] formulated the conjecture that for every Fttng class F, there exsts a normal Fttng class X such that F = F X. Later, n ths case, we say that F satsfes Lockett conjecture n S, where S s the class of all soluble groups. AMS Mathematcs Subject Classfcaton (2000): 20D10 Keywords: Fttng class; π-soluble group; Lockett class; Lockett par; Lockett conjecture. 1

2 About the Lockett conjecture, Bryce and Cossey [6] proved that Lockett conjecture holds for all soluble S-closed local Fttng classes and that every soluble Fttng class F satsfes Lockett conjecture f and only f F = F S, where S s the smallest normal Fttng class. In the paper [6], Bryce and Cossey also gave the concept of Lockett par (F, H) (see [6, 5.1]). They call an ordered par (F, H) of two Fttng classes F and H a Lockett par (or n brevty, an L-par) f F H = (F H). It s easy to see that f F s a Lockett class, F H and (F, H) s a Lockett par, then F satsfes Lockett conjecture n H. In partcular, f F s a Lockett class, F S, H = S and (F, H) s a Lockett par, then F satsfes Lockett conjecture n S. We say that F s an L H -class f F satsfes Lockett conjecture n H. If F satsfes Lockett conjecture n S, then F s called an L-class. Recall that a Fttng class F s sad to be S-closed f every subgroup of G F s n F. Bryce and Cossey [6] n the unverse S proved the exstence of Lockett pars. They showed that f F and H are S-closed Fttng classes, then (F, H) s a Lockett par. In connecton wth ths, the followng problem arses. Problem 1.1. Whch Fttng classes F and H satsfy that (F, H) s a Lockett par? In partcular, whch Fttng classes F satsfes Lockett conjecture n H? Note that, up to now, the problem was resolved only n the followng specal cases: 1) F {XN, XS π }, where X s some nonempty soluble Fttng class, H s FS π -njector and L p (F)-njector closed for all prmes p (see Bedleman and Hauck [1]); 2) F = S π and H S π (see Brson [4]); 3) F s an arbtrary soluble local Fttng class, and H s ψ-njector closed, where ψ s a local functon of F (see Vorob ev [15]); 4) F = S, and H = E s the class of all fnte groups (see Berger [3]); 5) H s a Fscher class(.e., H s a Fttng class whch s closed under takng subgroups of the form P N where P s a Sylow subgroup and N s a normal subgroup) or s closed under takng F- subgroups, whose ntersecton wth F-radcal of G s a normal subgroup of G, F satsfes the property that XE p F XE p E p for some Fttng class X and all p Char(X) (see Gallego [9]); 6) F s ω-local wth char(f) ω and H = E s the class of all fnte groups (see [12]). In ths paper, we wll construct a new famly of Lockett pars n the class S π of all fnte π-soluble groups. In order to acheve the purpose, n secton 3, we wll gve the concept of π-hr-closed Fttng class for some Fttng class X, whch, n fact, s a generalzed S-closed Fttng class. Base on ths, n secton 4, we obtan a famly of Lockett pars and also gve a new characterstc of the valdty of Lockett conjecture. As applcaton, some known results n [1], [3], [4], [9], [12] and [15] are obtaned as corollares of our results. Throughout ths paper, all groups are fnte. All unexplaned notaton and termnology are standard. The reader s referred to [8] and [10] f necessary. 2

3 2 Prelmnares Recall that a class F of groups s called a Fttng class provded the followng two condtons are satsfed: () f G F and N G, then N F. () f N 1, N 2 G and N 1, N 2 F, then N 1 N 2 F. From the condton () n the defnton, we see that, for every non-empty Fttng class F, every group G has a largest normal F-subgroup whch s called the F-radcal of G and denote by G F. The product FH of two Fttng classes F and H s the class (G : G/G F H). It s well known that the product of any two Fttng classes s also a Fttng class and the multplcaton of Fttng classes satsfes assocatve law (see [8, IX.1.12]). Recall that a class H of groups s sad to be a saturated homomorph f the followng condtons hold: 1) H s closed about homomorphc mages, that s, f G H and N G, then G/N H; 2) If G/Φ(G) H, then G H. A nonempty Fttng class H s sad to be a Fscher class f H H whenever K G H and H/K s a nlpotent subgroup of G/K (see [8, IX.3.3]). Obvously, any S-closed Fttng class s a Fscher class. We here cte some propertes of the operators and, whch are used n later proof. Lemma 2.1([13] and [8, X]). Let F and H be two non-empty Fttng classes. Then: a) If F H, then F H and F H ; b) (F ) = F = (F ) F F = (F ) = (F ) ; c) F F A, where A s the class of all abelan groups; d) If {F I} s a set of Fttng classes, then ( I F ) = I F. e) If H s a saturated homomorph, then (FH) = F H. f) If F s a homomorph (n partcular, a formaton) or a Fscher class, then F s a Lockett class. Suppose that G s a group, X s a class of groups and P s the set of all prmes. Then we let σ(g) = {p P : p G }, σ(x) = {σ(g) : G X} and Char(X) = {p P : Z p X}. Lemma 2.2 [8, X.1.20]. Char(F ) = Char(F) and σ(f ) = σ(f) for every Fttng class F. Let ω P. We denote by E ω the class of all fnte ω-groups, N denotes the class of all fnte nlpotent groups, S denotes the class of all fnte soluble groups. For a class F of groups, put F ω = F E ω. Followng [14], a map f : ω {ω } {Fttng class} s sad to be a ω-local Hatley functon (or n brevty, a ω-local H-functon). Then LR ω (f) denotes the Fttng class ( p π2 E p ) ( p π1 f(p)n p E p ) E ω f(ω ), where π 1 = Supp(f) ω, π 2 = ω\π 1. Here, Supp(f) := {a ω {ω} : f(a) } s called the support of the ω-local H-functon f. A Fttng class F s sad to be a ω-local f there exsts an ω-local H-functon f such that F = LR ω (f). If ω = P, then the ω-local Fttng class F s sad to be local. 3

4 Lemma 2.3[17, Theorem]. If F s an ω-local Fttng class and a Lockett class as well, then F = LR ω (F ) and F (a) s a Lockett class for all a ω {ω }. Moreover, F (p)n p = F (p) F F (p)e p, for all p ω. For constructng Lockett pars, we need the concept of the normal subgroup N(G) of G and ts propertes whch gven by Gallego [9]. We use H sn G to denote that H s a subnormal subgroup of G. A subnormal embeddng of a subnormal subgroup S of G s a monomorphsm α : S G such that Sα sn G. Let Snemb(S G) denote the set of all subnormal embeddng of S n G. Then let N(G) = x 1 x α : x S sn G and α Snemb(S G). Lemma 2.4 [9, Proposton (3.1)(3.2)]. Let G be a group and F a Fttng class. Then () N(G) s a characterstc subgroup of G. () N(G) G F for all G F. Lemma 2.5 [9, Proposton (4.1)]. Suppose that F, H and X are Fttng classes. Then the followng statements are equvalent: (a) F H X. (b) N(G) G H G X, for all G F. Lemma 2.6. (1) (Čunhn [7]) Every π-soluble group G has a Hall π-subgroup and any two Hall π-subgroups of G are conjugate n G. (2) [11] If F s a Fttng class, then K π (F) = (G S π : Hall π (G) F) s a Fttng class. We use Hall π (G) to denote the set of all Hall π-subgroups of a π-soluble group G. In ths connecton, we have the followng generalzed result of [9, Proposton 4.3]. Lemma 2.7. Let G be a group, H Hall π (G), H N and F a Fttng class. Then H N(G) N(HG F ). Proof. Let H 0 = x 1 x α : x S sn G, α Snemb(S G) and x, x α H. We frst prove that H N(G) = H 0. By the defnton of N(G), every generator of N(G) has the form g 1 g α, where g S sn G and α Snemb(S G). Let g = xy, where x, y g S such that x s a π-element and y s a π -element. Snce H Hall π (G), there exst elements a and b of G such that x a H and (x α ) b H. It s clear that g 1 g α O π (G)G = x 1 x α O π (G)G = (x a ) 1 (x α ) b O π (G)G. Note that S a sn G, (S α ) b sn G and there exsts an somorphsm from S a onto (S α ) b such that the mage of x α s (x α ) b. Therefore (x a ) 1 (x α ) b H 0 and so g 1 g α H 0 O π (G)G. Snce N(G) s generated by such elements g 1 g α, we have N(G) H 0 O π (G)G and hence H N(G) H 0 (H O π (G)G ). Snce O π (G)G /G s a π -group, H O π (G)G = H G. By [2, 21.3(2)], H G H 0. Hence H N(G) H 0. On the other hand, obvously, H 0 H N(G). Therefore H N(G) = H 0. Ths shows that H N(G) s generated by the elements x 1 x α, where x S sn G, α Snemb(S G) and x, x α H. Note that the subgroup x S F = x (S G F ) = S x G F s subnormal n 4

5 HG F. Analogously, ( x S F ) α = x α SF α H N(G) N(HG F ). s subnormal n HG F. Therefore x 1 x α N(HG F ) and 3 HR-classes In order to construct a new famly of Lockett pars, n ths secton, we wll defne the followng generalzed S-closed Fttng classes. Defnton 3.1. Suppose that π P and X s a Fttng class. (a) A subgroup T of G s called a π-hr-subgroup f T = HG X for some Hall π-subgroup H of G. (b) A Fttng class F s called π-hr-closed f every π-hr-subgroup of G belongs to F whenever G F. If F s σ-hr-closed for any σ P, then F s called an HR-class. (c) If X = (1), we wrte π-h-class nstead π-hr-class, and wrte H-class nstead HRclass. The followng examples show that the famly of the Fttng classes defned n Defnton 3.1 s wde. Examples 3.2. (1) Suppose that F s a S-closed Fttng class. Then, obvously, F s an π-hg X - class, for any nonempty Fttng class X. (2) Recall that for a Fttng class F and a group G F, f H F for every H Hall π (G), then F s sad to be π-hall closed [4]. Obvously, a Fttng class F s a π-h-class f and only f t s π-hall closed. (3) Let π = P and S be the smallest normal Fttng class. By the result n [6], S s an H-class. (4) For any set π P and any Fttng class H, the Fttng class K π (H) = (G S π : Hall π (G) H) was defned n [11] (see Lemma 2.6(2)). Obvously, K π (H) s π-h-closed f and only f H K π (H). Moreover, by the proof of [5, Proposton 4.4], we can see that for any τ π, the Fttng class K τ (H)Z s π-h-closed for any π-h-closed Fttng class Z. 4 On problem of the constructon of L-pars and L-classes In ths secton, we construct a famly of Lockett pars and gve a new characterstc of the valdty of Lockett conjecture. Defnton 4.1. Let F and H be two Fttng classes. () We say that F and H satsfy Property (α σ ) f σ π and there exsts a Fttng class X such that XS σ F XS σ S π σ, H K σ (N) and H s a σ-hr-class. () Let Char(F) be the characterstc of F and Char(F) = I σ, where σ and σ σ j = for all, j I( j). We say that F and H satsfy Property (α) f F and H satsfy Property (α σ ) for all I. Lemma 4.2. Let F and H be Fttng classes. If F and H satsfes Property (α σ ), then F H (F H) S π σ. 5

6 Proof. Let (F H) S π σ = M. In order to prove F H M, by Lemma 2.5, we only need to prove that N(G) G F G M for every G H. Suppose that G H and H Hall σ (G). Then HG X /G X Hall σ (G/G X ) and so HG X /G X S σ. Hence, from [8, IX, 1.11], we have H XS σ F. Snce H s a σ-hr-class by hypothess, HG X H. Hence, HG X F H. Besdes, snce H s a Hall σ-subgroup, by Lemma 2.7, we have H N(G) N(HG X ). Hence, H N(G) G F N(HG X ) G F. We clam that N(HG X ) G F (HG X G F ) (F H). In fact, snce HG X F H, by Lemma 2.4 N(HG X ) (HG X ) (F H) and so N(HG X ) G F (HG X ) (F H) G F. But, because HG X G F HG X, by [8, IX.1.1((a)], we have (HG X G F ) (F H) = (HG X ) (F H) (HG X G F ) = (HG X ) (F H) G F. Hence N(HG X ) G F (HG X ) (F H) G F = (HG X G F ) (F H). Now we prove HG X G F G. In fact, snce G F XS σ S π σ, G F /G XSσ S π σ. Then, by theorem [8, IX.1.12], (G F /G X )/(G XSσ /G X ) = (G F /G X )/(G/G X ) Sσ S π σ and so G XSσ /G X Hall σ (G F /G X ). On the other hand, snce H Hall σ (G), (H G F )G X /G X Hall σ (G F /G X ). Hence G XSσ = (H G F )G X = HG X G F and HG X G F G. Ths mples that (HG X G F ) (F H) G (F H). Therefore, H N(G) G F G (F H) N(G) G F. But by Lemma 2.4(), N(G) G. Hence G (F H) (N(G) G F ) = (N(G) G F ) (F H) by [8, IX, 1.1 (a)]. It follows that H (N(G) G F ) (N(G) G F ) (F H). Snce H 1 := H (N(G) G F ) s a Hall σ-subgroup of N(G) G F, (N(G) G F ) : H 1 s a σ -number. But snce H 1 (N(G) G F ) (F H), (N(G) G F )/(N(G) G F ) (F H) S π σ. Hence N(G) G F (F H) S π σ = M. Consequently N(G) G F G M. Ths completes that proof. The followng theorem descrbes a new and wde famly of Lockett pars. In partcular, the theorem gve some new Fttng classes whch satsfy Lockett conjecture. Theorem 4.3. Let F and H be two Fttng classes. If F and H satsfy Property (α), then (F, H) s an L-par. In partcularly, f F H, then F s an L H -class, that s, F satsfes Lockett conjecture n H. Proof. By Lemma 4.2, we only need to prove that f F H (F H)S π σ for every I, then F H = (F H). Frstly, by Lemma 2.1, we have (F H) F H. Conversely, assume that t s not true and let G be a group n F H \ (F H) of mnmal order. Then G has a unque maxmal normal subgroup M = G (F H). Snce G F H, G F H by Lemma 2.1(a)(b). Then by usng Lemma 2.1(b)(c), we obtan that G (F H) A, where A s the class of all abelan groups. Hence, G/M has a unque maxmal normal subgroup of order p and so G/M Z p. Snce G F H, p Char(F H) by [8, Lemma IX.1.7]. It follows from Lemma 2.2 that there exsts σ 0 Char(F H) for 0 I such that p σ 0 Char((F H) ). Therefore G/M S σ0. On the other hand, snce σ, F and H satsfy condtons of Lemma 4.2, F H (F H) S π σ. 0 Snce G F H and M = G (F H), we have G/M S π σ. Ths mples that G = M (F H). 0 Ths contradcton shows that (F, H) s an L-par. 6

7 Now assume that F H. In order to prove that F s an L-class (that s, F satsfes Lockett conjecture), clearly, we only need to prove that F s a Lockett class (that s, F = F). By the hypothess, F satsfes Property (α). Hence XS σ F XS σ S π σ for all I. Then by Lemma 2.1(a), we see that F (XS σ S π σ ). But by [15, Corollary], XS σ S π σ s local and so t s a Lockett class by [15, Lemma 5]. Therefore, (XS σ S π σ ) = XS σ S π σ and thereby F FS π σ for all I. By Lemma 2.2, Char(F) = Char(F ). Now analogous proof from F H (F H) S π σ to F H (F H), we obtan that F = F. Ths completes the proof. 5 Applcatons By usng Theorem 4.3, we mmedately obtan the followng known results about descrpton of L-par and L-class. Corollary 5.1(Bryce, Cossey [6]). If F and H are soluble S-closed Fttng classes, then (F, H) s an L-par. In partcularly, (F, S) s an L-par, that s, every S-closed Fttng class s an L-class. Proof. By [17, Theorem], F and H are local Fttng classes. Put σ = {p} n Theorem 4.3, for all p Char(F) and I. Snce H s S-closed, H s an HR-class for any Fttng class X. Besdes, by Lemma 2.3, for ω = P and every p Char(F), we have F (p)n p F F (p)n p S p. Thus, by Theorem 4.3, (F, H) s an L-par. Besdes, by Lemma 2.1(f), F = F. If H = S, then F s an L-class, that s, F satsfes Lockett conjecture. Corollary 5.2[see 12, Theorem B]. If F = LR ω (F ) s ω-local Fttng class wth Char(F) ω, then (F, S π ) s an L-par and F s an L-class. Proof. By Lemma 2.3, F (p)n p F F (p)n p S π p for all p Char(F). Hence, f put σ = {p} for all p Char(F) and I and let H = S π, the F and H satsfy the hypothess of Theorem 4.3. Thus, by Theorem 4.3, (F, S π ) s an L-par and so, clearly, F s an L-class snce F H. Put ω = π, then by Corollary 5.2, we have Corollary 5.3. If F s a local Fttng class, then (F, S π ) s an L-par. Put ω = π = P, then by Corollary 5.2, we obtan Corollary 5.4[15]. Lockett conjecture holds for every soluble local Fttng class F, that s, f F S, then par (F, S) s an L-par. Corollary 5.5[1]. Let F {XN, XS π S π }, where X s some nonempty soluble Fttng class. Then F satsfes Lockett conjecture. Proof. By [15, Corollary], XN and XS π S π are all local Fttng classes. Hence by Corollary 5.3, the statement holds. Corollary 5.6. Let σ π and F, H be Fttng classes such that F = S σ H. Then (F, H) s an L-par and S σ s an L H -class. Proof. Obvously, F s local. By Lemma 2.3, for ω = P, F satsfes the related condtons of Theorem 4.3 for F f put σ = {p} for all p Char(F) and I. Besdes, by [15, Lemma 5], F s a Lockett class, that s, F = F. Now we prove that (F, H) s an L-par, that s, F H = (F H). 7

8 If σ = φ, then F = (1) and so t s trval. If σ = {p}, then by [8, X.1.23], (N p ) = N p and so N p H = (N p H) = (N p ). Put σ 2. Snce S σ H, then H s a p-h-class and H K p (N) for every p Char(F) = σ. Thus by Theorem 4.3, (S σ, H) s an L-par. For π = P we have Corollary 5.7[4]. Let F, H be soluble Fttng classes and σ P. If F = S σ H, then (F, H) s an L-par and F s an L H -class. References [1] Bedleman J. C. and Hauck P., Über Fttngklassen and de Lockett-Vermutung, Math Z., 167, 1979, [2] Belonogov V.A., Problems on Theory of Groups, Moscow, Nauka, [3] Berger T.R., Normal Fttng pars and Lockett s conjecture, Math Z., 163, 1978, [4] Brson O.Y., Hall operaters for Fttng class, Arch. Math.(Basel), 33, 979/1980, 1-9. [5] Brson O.Y., Hall-closure and products of Fttng classes, J. Austral Math. Soc.(Seres A) 32, 1982, [6] Bryce R.A. and Cossey J., A problem n the theory of normal Fttng classes, Math. Z., 141, 1975, [7] Čunhn S. A., On π-separable groups, Doklady Akad, Nauk SSSR (N. S.), 59, 1948, [8] Doerk K. and Hawkes T.O., Fnte Soluble Groups, Walter de Gruyter, Berln, New York, [9] Gallego M.P., Fttng pars from drect lmtes and the Lockett conjecture, Comm. n Algebra, 24(6), 1996, [10] Guo W., The Theory of Classes of Groups, Scence Press-Kluwer Academc Publshers, Bejng- New York-Dordrecht-Boston-London, [11] Guo W., L B., On Shemetkov Problem for Fttng classes, Contrbutons to Algebra and Geometry, 48(1), 2007, [12] Guo W., Shum K. P. and Vorob ev N.T., Problems related to the Lockett conjecture on Fttng classes of fnte groups, Indag. Mathem., N.S., 19(3), 2008, [13] Lockett P., The Fttng class F, Math Z., 137, 1974, [14] Shemetkov L.A. and Skba A.N., Multply ω-local formatons and Fttng classes of fnte groups, Sberan Advances n Mathematcs, 10(2), 1999,

9 [15] Vorob ev N.T., On radcal classes of fnte groups wth the Lockett condton, Math. Zametk, 43, 1988, ; translated n Math. Notes, 43, 1988, [16] Vorob ev N.T., Localty of solvable subgroup-closed Fttng classes, Math. Zametk, 51, 1992, 3-8; translaton n Math, Notes 51, 1992, [17] Vorob ev N.T., On largest ntegrated Hartley s functon, Proc. Gomel Unversty, Problems n Algebra, 1(15), 2002,