Journal of Algebra 358 (2012) Contents lists available at SciVerse ScienceDirect. Journal of Algebra.

Transcription

1 Journal of Algebra 358 ( Contents lsts avalable at ScVerse ScenceDrect Journal of Algebra A characterzaton of domnant local Fttng classes P. Hauck a,,1,v.n.zahursky b,2 a Wlhelm-Schckard-Insttut für Informatk, Unverstät Tübngen, Sand 13, Tübngen, Germany b Department of Mathematcs, Vtebsk State Technologcal Unversty, Moscow Avenue 72, Vtebsk, Belarus artcle nfo abstract Artcle hstory: Receved 28 October 2011 Avalable onlne 12 March 2012 Communcated by Gernot Stroth Keywords: Fnte groups Fttng classes Domnant Fttng classes Local Fttng classes Inectors Radcals A Fttng class F s called domnant n the class of all fnte soluble groups S f F S and for every group G S any two F-maxmal subgroups of G contanng the F-radcal G F of G are conugate n G. In ths paper a characterzaton of domnant local Fttng classes n the class of all fnte soluble groups s establshed Elsever Inc. All rghts reserved. 1. Introducton In the theory of classes of fnte soluble groups, a basc result whch generalzes fundamental theorems of Sylow and Hall s the theorem of Fscher, Gaschütz and Hartley 1 on exstence and conugacy of F-nectors n fnte soluble groups G for every Fttng class F. Recall that a subgroup V of a group G s called an F-nector of G f V N s an F-maxmal subgroup of N for every subnormal subgroup N of G 1,2. There has been substantal research on characterzatons of F-nectors for varous types of Fttng classes F (e.g For domnant Fttng classes, by ther very defnton, there s a convenent descrpton of nectors by means of radcals. Namely, a Fttng class F s sad to be domnant n the class of all soluble groups S f F S and f for every group G S any two F-maxmal subgroups of G contanng the F-radcal G F of G are conugate n G. These are then the F-nectors of G. * Correspondng author. E-mal addresses: hauck@nformatk.un-tuebngen.de (P. Hauck, zagursk@yandex.ru (V.N. Zahursky. 1 Research supported by Proyecto MTM C03-02, Mnstero de Cenca e Innovacón, Span. 2 Research supported by a Research Grant for Doctoral Canddates and Young Academcs and Scentsts, DAAD (German Academc Exchange Servce /$ see front matter 2012 Elsever Inc. All rghts reserved. do: /.algebra

2 28 P. Hauck, V.N. Zahursky / Journal of Algebra 358 ( In 1971 Lockett 5 proved that f F s a soluble domnant Fttng class then ether the class N of fnte nlpotent groups s contaned n F or F =, the class of all soluble π -groups for some π P. The classes F = are very smple examples of so-called local Fttng classes. Frst consdered by Hartley 4, local Fttng classes have been ntroduced explctely and nvestgated by D Arcy 7 n an attempt to dualze the concept of local defnton for saturated formatons. Local Fttng classes consttute a large famly of Fttng classes, though not every Fttng class s local 2, IX.3.7. Apart from F = also many known domnant Fttng classes F contanng N are local. Examples of such Fttng classes are: the class N (Fscher 3 and, more generally, the classes XN for any nonempty Fttng class X (Hartley 4, the classes N π of π -nlpotent groups and the classes of π - closed groups for all π P. All of these examples belong to a partcular type of local Fttng classes, called Hartley classes. In 2008 W. Guo and N.T. Vorob ev 8 proved a general result statng that all Hartley classes are domnant. However, not every local Fttng class s a Hartley class. In fact, there are local Fttng classes F contanng N whch are not domnant 2, IX.4.4. Therefore, t s an nterestng problem to decde whch local Fttng classes are domnant. The man result of ths paper settles ths queston and gves a complete characterzaton of domnant local Fttng classes contanng N. 2. Prelmnares All groups consdered n ths paper are fnte and soluble. We denote by Z n the cyclc group of order n. For notaton and basc results on classes of groups we refer to 2. A local method to construct Fttng classes has been ntroduced by Hartley 4 and D Arcy 7: A map f whch assgns to each prme p a Fttng class f (p s called a Hartley functon or an H-functon for short 9. A Fttng class F s called local 7,9, f F = p π f (ps ps p, where f s an H-functon and π = Char(F. Ifall f (p, p π, are contaned n F then f s called an ntegrated H-functon of F. Gven a partton {π : I} of P, a Fttng class H of full characterstc s called a Hartley class 4 f there s an H-functon h whch s constant on each π, h(π := h(p for all p π, such that H = h(π I. It s easy to show that every Hartley class s a local Fttng class (see also Corollary 1 below. We recall that a Fttng class F s called a Lockett class f (G H F = G F H F for all groups G and H. Lemma 1. (See Vorob ev 10, corollary. Every local Fttng class F s defned by a unque maxmal ntegrated H-functon F. It s characterzed among all ntegrated H-functons for F by the property that F (p = F (ps p s a Lockett class for all prmes p Char(F. Lemma 2. (See D Arcy 7. Let F be a local Fttng class wth maxmal ntegrated H-functon F. Let G be a group and Σ a Sylow system of G. Defne B(Σ = G p Σ C G p (G F /G F (p. (a 7, Lemma 11 B(Σ contans every F-subgroup H G F of G nto whch Σ reduces. (b 7, Lemma 10 If N s a subnormal subgroup of G, then B(Σ N = B(Σ N, the correspondng subgroup of N wth respect to the Sylow system Σ NofN. (c 7, Theorem 14 B(Σ F f and only f all maxmal F-subgroups contanng G F are F-nectors. Lemma 3. Let F be a Lockett class, H a group and p a prme number. If H Z p FS p,thenh Z p F.

3 P. Hauck, V.N. Zahursky / Journal of Algebra 358 ( Proof. Assume that H / F. Then by 2, X.2.1 (H Z p F = (H F H, where H s the base group of the regular wreath product H Z p. Therefore p dvdes (H Z p /(H Z p F, contradctng (H Z p FS p. Thus H F and consequently H Z p FS p FS p = F. 3. The results The followng theorem characterzes domnant local Fttng classes F N n terms of ther maxmal ntegrated H-functons F. Theorem. Let F be a local Fttng class of full characterstc wth maxmal ntegrated H-functon F. Then the followng two statements are equvalent: ( F s domnant. ( (F (p F (qs {p,q} = F (p F (q or F (p F (q = r P F (r for every par of dstnct prme numbers pandq. Proof. ( (: Suppose that condton ( does not hold. Then there exst dstnct prme numbers p, q, r such that (F (p F (qs p F (q and F (p F (q F (r. (Note that because of solublty, (F (p F (qs {p,q} = F (p F (q s equvalent to (F (p F (qs p = F (p F (q and (F (p F (qs q = F (p F (q. (1 There s a group H such that H (F (p F (q \ F (r and H Z p / F (q: By 2, X.2.4 and X.2.13 there exsts a group H 0 F (p F (q wth H 0 Z p / F (q. Moreover, there exsts a group H 1 (F (p F (q \ F (r. IfH 1 Z p / F (q, thenh = H 1 satsfes (1. So we may assume that H 1 Z p F (q. ThenH 0 H 1 (F (p F (q \ F (r. Applcaton of the quas-r 0 lemma 2, IX.1.13 yelds (H 0 H 1 Z p / F (q. ThusH = H 0 H 1 satsfes (1. (2 Let G = ((H Z r Z p Z q and H = ((H the drect product of r p q copes of H nsde the terated wreath product G. ThenG F = H: As F (r s a Lockett class and H / F (r, Lemma3yeldsH Z r / F F (rs r.snceh F(p F, t follows therefore that (H Z r F = H. Applyng 2, X.2.1(a twce yelds the asserton of (2. (3 Fnal contradcton: Let P Syl p (G and Q Syl q (G. Then HP F (psp = F (p F and HQ F (qsq = F (q F. Snce F s domnant, t follows from (2 that there exsts an F-nector V of G such that HP V and HQ g V for some g G. Note that V ((H Z r s subnormal n G and hence contaned n G F = H by (2. Therefore V / H s a Hall-{p, q}-subgroup of G/ H. It follows that V = (H r Z p Z q where H r stands for the drect product of r copes of H. Consequently,(H r Z p Z q F F (qs q. F (q s a Lockett class and thus Lemma 3 yelds H r Z p F (q whence H Z p F (q by 2, X.2.4. Ths contradcts (1. ( (: Suppose that F s not domnant. Let G be a group of mnmal order wth respect to the property that not every F-maxmal subgroup of G contanng G F s an F-nector of G. Clearly, G / F. LetN G F be a maxmal normal subgroup of G. Then G : N =p for some prme p. Let Σ be a Sylow system of G and B = B(Σ as n Lemma 2. (1 Let V = B N. ThenB / F and V = B F : By Lemma 2(c, B / F. Moreover by Lemma 2(b, V = B N = B(Σ N s the correspondng subgroup of N for the Sylow system Σ N of N. As N < G, another applcaton of Lemma 2(c yelds V F. HenceV = B F. (2 G F /G F (p Z(G/G F (p : Assume that C = C G (G F /G F (p s a proper subgroup of G. ThenC s not a counterexample. Snce C s normal n G, t follows from Lemma 2(b, (c that B C F. SnceB C s normal n B, (1yelds B C B F = V.ForG p Σ we have C G p (G F /G F (p Syl p (B. Nowsnce B/V =p, t follows that B = VC G p (G F /G F (p V (B C = V, a contradcton. (3 Let G p be the Sylow p-subgroup of G contaned n Σ and W = G p G F.ThenW F, W B, B = VW and G p W F (p :

4 30 P. Hauck, V.N. Zahursky / Journal of Algebra 358 ( G p G F (p F (ps p F. SnceG F /G F (p Z(G/G F (p by (2, W s the product of the two normal F-subgroups G F and G p G F (p.thusw F. By Lemma 2(a, W B. SnceF F (ps p, t follows that G p W F (p.moreover, B/V =p mples B = VW. (4 Let G F = V 0 < V 1 < < V r = V < B = V r+1 be a seres of subgroups such that V s maxmal among the proper W -nvarant normal subgroups of V +1 for all = 0,...,r. Clearly, all V +1 /V are elementary abelan. Snce G F W = W F and VW = B / F, there s a maxmal number, such that V W F and V +1 W / F. Clearly, < r. SetX = V and Y = V +1. Then XW = X(XW F (p, Y /X =q a for a prme q p and Y = XY F (q : If Y /X s a p-group, then Y W = XW as W contans a Sylow p-subgroup of G reducng nto Y. But ths contradcts the choce of X and Y.Consequently, Y /X =q a for a prme q p. Bydefnton of W, XW/X = XG p /X s a p-group. On the other hand, snce XW F F (ps p, XW/(XW F (p s a p -group. It follows that XW = X(XW F (p. Fnally, Y F F (qs q and Y /X =q a mply Y = XY F (q. (5 (F (p F (qs {p,q} = F (p F (q: Set D = r P F (r. If (5 does not hold, then by hypothess (, F (p F (q = D. LetG q be the Sylow q-subgroup of G contaned n Σ. SnceV q = V G q C Gq (G F /G F (q Syl q (B, tthenfollows from (2 that V q C G (G F /G D.NowDN p P F (ps ps p = F mples V q C G (F (G/G D. Therefore V q G D /G D C G/GD (F (G/G D F (G/G D G F /G D. Snce V q G D /G D Syl q (V /G D, t follows that V /G F s a q -group, contradctng (4. (6 Y F (q,(xw F (p, X F (q X F (p X F (q : BythechoceofX and Y n (4, X YW and hence X F (p, X F (q YW.Now, (XWF (p, X F (q (XWF (p X F (q = (XW F (p X X F (q = X F (p X F (q. Consequently, X F (q, Y F (q,(xw F (p X F (q,(xw F (p X F (p X F (q and (XWF (p, X F (q, Y F (q X F (q X F (p, Y F (q X F (p X F (q. Now statement (6 follows from the Three Subgroups Lemma 2, A.7.6. (7 Y F (q,(xw F (p X F (q = Y F (q : Y,(XW F (p s normal n YW. Therefore, by the choce of X and Y, Y,(XW F (p X equals X or Y. Suppose, Y,(XW F (p X. Then YW s the product of the two normal F-subgroups Y and XW = X(XW F (p (cf. (4. Hence YW F, contradctng the choce of Y. It follows that Y,(XW F (p X = Y. By (4, Y = XY F (q. Thus Y F (q,(xw F (p X =Y,(XW F (p X = Y. Snce Y F (q,(xw F (p Y F (q, t follows that Y F (q = Y F (q,(xw F (p X Y F (q = Y F (q,(xw F (p (X Y F (q = Y F (q,(xw F (p X F (q. (8 Y = XY F (p : Let D = X F (p F (q. By (6 and (7, Y F (q /D s the central product of Y F (q,(xw F (p D/D and X F (q /D. LetQ Syl q (Y F (q,(xw F (p. AsY F (q /X F (q s a q-group, Y F (q = QX F (q and QD s a normal subgroup of Y F (q. By (5, QD F (p F (q and therefore QD Y F (p F (q Y F (p. It follows that Y F (q Y F (p X F (q. Accordng to (4, Y = XY F (q, and (8 follows.

5 P. Hauck, V.N. Zahursky / Journal of Algebra 358 ( (9 Fnal contradcton: Note that X, Y F (p (XW F (p X,(XWF (p Y F (p X F (p Y F (p = Y F (p. Thus X normalzes Y F (p (XW F (p. Now Y F (p (XW F (p /Y F (p = (XW F (p /((XW F (p Y F (p s a dvsor of (XW F (p /X F (p = X(XW F (p /X and hence s a p-power. Consequently, Y F (p (XW F (p F (ps p F. Hence,usng (4 and (8, YW s the product of the normal F-subgroups X and Y F (p (XW F (p. It follows that YW F, contradctng the choce of Y n (4. Remark. It s easy to see that condton ( of the theorem s equvalent to the followng statement: If π P, then( p π F (p = p π F (p or p π F (p = r P F (r. Corollary 1. (See Guo and Vorob ev 8, theorem. Every Hartley class s domnant. Proof. Let H = h(π I be a Hartley class for a partton {π : I} of the set P of all prmes. By 2, X.1.27.b, we may assume that all h(π are Lockett classes. For p π we set F (p = h(π h(π. By 2, X.1.26.b, each F (p s a Lockett class. It s obvous that F (p = F (ps p H for all p P. Now F (ps p S p = p P I p π = I = I = ( I I ( h(π ( h(π ( h(π = I = H. S p S p N π Therefore the Fttng class H s local wth maxmal ntegrated H-functon F (cf. Lemma 1. It s easy to see that (F (p F (qs {p,q} = F (p F (q f p and q are n the same set π of the partton of P and F (p F (q = r P F (r otherwse. Hence H s domnant by the theorem. Corollary 2. (See D Arcy 7, Corollary 17. Let {π : I} be a partton of P and let π be a set of prmes. Then for any Fttng class X, the local Fttng classes X( and X are domnant. Proof. It s easy to check that F = X( s a local Fttng class wth maxmal local ntegrated H-functon F (p = X for p π (cf. Lemma 1. Domnance of F follows now from the theorem. The asserton about X = X S s an mmedate consequence of Corollary 1. Examples of local Fttng classes contanng N, whch are not domnant, are gven n the followng corollary.

6 32 P. Hauck, V.N. Zahursky / Journal of Algebra 358 ( Corollary 3. (Cf. 2, IX.4.4. N π s a local Fttng class whch s not domnant for every proper subset π of P of sze at least 2. Proof. The Fttng class N π s local and t s defned by the greatest ntegrated H-functon F gven by F (p = S p for p π and F (p = N π for p π.thenf(p F (q = S p f p π and q / π.as π 2, r π F (r = (1, and t follows from the theorem that N π s not domnant. References 1 B. Fscher, W. Gaschütz, B. Hartley, Inektoren endlcher auflösbarer Gruppen, Math. Z. 102 ( K. Doerk, T.O. Hawkes, Fnte Soluble Groups, Walter de Gruyter, Berln/New York, B. Fscher, Klassen konugerter Untergruppen n endlchen auflösbaren Gruppen, Habltatonsschrft, Unverstät Frankfurt (M, B. Hartley, On Fscher s dualzaton of formaton theory, Proc. London Math. Soc. (3 19 ( P. Lockett, On the theory of Fttng classes of fnte soluble groups, PhD thess, Unversty of Warwck, A. Mann, Inectors and normal subgroups of fnte groups, Israel J. Math. 9 ( P. D Arcy, Locally defned Fttng classes, J. Austral. Math. Soc. (Ser. A 20 ( W. Guo, N.T. Vorob ev, On nectors of fnte soluble groups, Comm. Algebra 36 ( N.T. Vorob ev, On a conecture of Hawkes for radcal classes, Sberan Math. J. 37 ( N.T. Vorob ev, On the largest ntegrated Hartley functon, Proc. Gomel State Unversty 1 (15 ( (n Russan, Englsh summary.