SE-supplemented subgroups of finite groups

SE-Supplemented Subgroups of Finite Groups

WENBINGUO(*) - ALEXANDERN. SKIBA(**) - NANYINGYANG(*)

ABSTRACT- LetGbe a finite group,Ha subgroup ofGandHseGbe the subgroup generated by all subgroups ofH which areS-quasinormally embedded inG.

Then we say thatH isSE-supplemented inGifGhas a subgroupT such that HT ˆGandH\THseG. We investigate the influence ofSE-quasinormally embedded of some subgroups on the structure of finite groups. Our results improve and extend a series of recent results.

MATHEMATICSSUBJECTCLASSIFICATION(2010). 20D10, 20D20, 20D25.

KEYWORDS.S-quasinormal subgroup,S-quasinormally embedded subgroup, FE- supplemented subgroup, formation.

1. Introduction

Throughout this paper, all groups are finite and G denotes a finite group.

An interesting question in theory of finite groups is to determine the influence of the embedding properties of members of some distinguished families of subgroups of a group on the structure of the group.

Recall that a subgroupHofGis said to beS-quasinormal,S-permu- table, or p(G)-permutable in G (Kegel [22]) if HPˆPH for all Sylow subgroupsPofG. A subgroupHofGis said to beS-quasinormally em- beddedorS-permutably embeddedinG(Ballester-Bolinches and Pedraza- Aguilera [6]) if each Sylow subgroup ofHis also a Sylow subgroup of some

(*) Indirizzo degli A.: School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, P. R. China

E-mail: wbguo@ustc.edu.cn yangny@ustc.edu.cn

(**) Indirizzo dell'A.: Department of Mathematics, Francisk Skorina Gomel State University, Gomel 246019, Belarus

E-mail: alexander.skiba49@gmail.com

S-quasinormal subgroup ofG. A subgroupAofGis said to be c-normal (Wang [32]) (c-supplemented(Ballester-Bolinches, Wang and Guo [9])) in G if Ghas a normal subgroup (a subgroup, respectively) T such that ATˆGandA\TA_G.

LetF be a class of groups. If 12 F, then we writeG^F to denote the intersection of all normal subgroupsNofGwithG=N2 F. The classFis said to be a formation if eitherF ˆ[or 12 F and every homomorphic image ofG=G^F belongs toFfor every groupG. A formationFis said to be saturatedorlocalifF contains every groupGwithG^F F(G). A classF of groups is said to be solubly saturatedorBaer-local(see [11, Chapter IV, Definition 4.9]) ifFcontains every groupGwithG^F F(N) for some soluble normal subgroupNofG.

Researches of many authors are connected with analysis of the fol- lowing general question: Let F be a saturated formation containing all supersoluble groups and G a group with a normal subgroup Esuch that G=E2 F. Under what conditions on Ethen, does G belong toF ?

We recall some recent results in this direction. If F is a saturated formation containing all supersoluble groups andGhas a normal subgroup Esuch thatG=E2 F, then the following results are true:

(1) If the cyclic subgroups of E of prime order or order 4 are or S- quasinormal (Ballester-Bolinches and Pedraza-Aguilera [7], Asaad and CsoÈrgoÈ [2]), orc-normal (Ballester-Bolinches and Wang [8]), or c-supple- mented (Ballester-Bolinches, Wang and Guo [9], Wang and Li [35]) inG, thenG2 F.

(2) If the cyclic subgroups ofF(E) of prime order or order 4 are orS- quasinormal (Li and Wang [24]), orc-normal (Wei, Wang and Li [36]), orS- quasinormally embedded (Li and Wang [25]), orc-supplemented (Wang, Wei and Li [34], Wei, Wang and Li [37]) inG, thenG2 F.

(3) If the maximal subgroups of every Sylow subgroup ofEare orS- quasinormal (Asaad [1]), orc-normal (Wei [38]), orc-supplemented (Bal- lester-Bolinches and Guo [5]) inG, thenG2 F.

(4) If the maximal subgroups of every Sylow subgroup ofF(E) are or S-quasinormal (Li and Wang [26]), orc-normal (Wei, Wang and Li [37]), or c-supplemented (Wei, Wang and Li [34]), or S-quasinormally embedded (Li and Wang [25]) inG, thenG2 F.

(5) IfEis soluble and the maximal subgroups of every Sylow subgroup of F(E) areS-quasinormally embedded inG, thenG2 F(Asaad and Heliel [3]).

In these results,F(E) is the generalized Fitting subgroup ofE, that is, the product of all normal quasinilpotent subgroups ofE[21, Chapter X].

Bearing in mind the above-mentioned results, it is natural to ask:

(I) Is it true that all the above-mentioned results can be strengthened by considering the more general case, whereFis a Baer-local formation?

(II) Is it true that all the above-mentioned results can be improved by using some weaker condition?

In this paper, we give positive answers to both these questions.

Let H_seGbe the subgroup generated by all the subgroups of Hwhich areS-quasinormally embedded inG. We callH_seGtheSE-coreofHinG.

DEFINITION1.1. Let H be a subgroup of G. We say that H is SE-sup- plemented in G if there exists some subgroup T of G such that HTˆG and H\TH_seG.

The key to solving Questions (I) and (II) are the following two results.

THEOREM1.2. Let Ebe a normal subgroup of G. If the cyclic subgroups of Eof prime order or order4are SE-supplemented in G, then each chief factor of G below Eis cyclic.

THEOREM 1.3. Let Ebe a normal subgroup of G. If the maximal subgroups of every Sylow subgroup of Eare SE-supplemented in G, then each chief factor of G below Eis cyclic.

A chief factor H=K of G is called F-central in G provided (H=K)^j(G=C_G(H=K))2 F.

In [31], the following result was proved.

THEOREM 1.4 [31, Theorem 3.1]. Let F be any formation and Ea normal subgroup of G. If each chief factor of G below F(E)isF-central in G, then each chief factor of G below EisF-central in G as well.

Base on these theorems, we may directly obtained the following results.

COROLLARY 1.5. Let Ebe a normal subgroup of G. If the cyclic subgroups of F(E) of prime order or order 4 are SE-supplemented in G, then each chief factor of G below Eis cyclic.

COROLLARY 1.6. Let Ebe a normal subgroup of G. If the maximal subgroups of every Sylow subgroup of F(E)are SE-supplemented in G, then each chief factor of G below Eis cyclic.

It is clear that ifFis a Baer-local formation containing all supersoluble groups and G has a cyclic normal subgroupE such thatG=E2 F, then G2 F (see Lemma 2.17 below). Hence from Theorems 1.2, 1.3 and 1.4, we also directly get the following

THEOREM1.7. LetF be a Baer-local formation containing all super- soluble groups and G has normal subgroups XEsuch that G=E2 F.

Suppose that the cyclic subgroups of X of prime order or order4are SE- supplemented in G. If either XˆEor XˆF(E), then G2 F and each chief factor of G below Eis cyclic.

THEOREM 1.8. Let F be a Baer-local formation containing all su- persoluble groups and G has normal subgroups XEsuch that G=E2 F.

Suppose that the maximal subgroups of every Sylow subgroup of X are SE-supplemented in G. If either XˆEor X ˆF(E), then G2 Fand each chief factor of G below Eis cyclic.

It is easy to see that all S-quasinormal subgroups,S-quasinormally embedded subgroups, c-normal subgroups, and c-supplemented sub- groups are all SE-supplemented in G. But the converse is not true (see the example in Section 5). Hence Theorems 1.7 and 1.8 give af- firmative answers to Questions (I) and (II) and consequently, a large number of known results (for example, the above results in (1)-(5)) are generalized.

All unexplained notations and terminologies are standard. The reader is referred to [11], [4] and [16] if necessary.

2. Preliminaries

LEMMA2.1 [22]. Let G be a group and HKG.

(1) If H is S-quasinormal in G, then H is S-quasinormal in K.

(2) Suppose that H is normal in G. Then K=H is S-quasinormal in G if and only if K is S-quasinormal in G.

(3) If H is S-quasinormal in G, then H is subnormal in G.

From Lemma 2.1 (3) we get

LEMMA2.2. If H is an S-quasinormal subgroup of G and H is a p-group for some prime p, then O^p(G)NG(H).

LEMMA2.3. Suppose that A, B are subgroups of G.

(1) If A is S-quasinormal in G, then A\B is S-quasinormal in B[10].

(2) If A and B are S-quasinormal in G, then A\B is S-quasinormal in G[22].

(3) If A is S-quasinormal in G, then A=A_Gis nilpotent[10].

LEMMA2.4 [6]. Let G be a group and HKG.

(1) If H is S-quasinormally embedded in G, then H is S-quasinor- mally embedded in K.

(2) If H is normal in G and Eis an S-quasinormally embedded subgroup of G, then EH is S-quasinormally embedded in G and EH=H is S-quasinormally embedded in G=H:

LEMMA 2.5. Suppose that H is an S-quasinormally embedded sub- group of G. If HO_p(G)for some prime p, then H is S-quasinormal in G.

PROOF. Suppose that H is a Sylow p-subgroup of some S-quasi- normal subgroup Eof G. ThenHˆOp(G)\E. Hence by Lemma 2.3(2), HisS-quasinormal inG.

LEMMA2.6. Suppose that N is a normal subgroup of G and HKG.

Then:

(1) H_seG/H.

(2) H_seGH_seK.

(3) H_seGN=N(HN=N)_se(G=N).

(4) If(jNj;jHj)ˆ1, then HseGN=Nˆ(HN=N)se(G=N). (5) (H_seG)^xˆ(H^x)_seG, for any x2G.

PROOF. Let Lbe an S-quasinormally embedded subgroup of Gcon- tained inH,qbe a prime dividingjLj,Qa Sylowq-subgroup ofLandEanS- quasinormal subgroup ofGsuch thatQ2Syl_q(E).

(1) Let x2H. Then L^xH. IfR is a Sylow q-subgroup of L^x, then RˆQ1xfor some Sylowq-subgroupQ1ofL. Without loss of generality, we may assume thatQ₁ˆQ. Obviously,Q^x2Syl_q(E^x) andE^xis anS-quasi- normal subgroup of G. Hence L^x is an S-quasinormally embedded sub- group ofG. This implies that (1) holds.

(2) By Lemma 2.3(1),E\Kis anS-quasinormal subgroup ofK. Since QE\K, Q is a Sylow q-subgroup ofE\K. Hence LHseK and so H_seGH_seK.

(3) ClearlyLN=NHN=NandLN=Nis anS-quasinormally embedded subgroup ofG=Nby Lemma 2.4(2). HenceH_seGN=N(HN=N)_se(G=N).

(4) In view of (3), we only need to prove that (HN=N)_se(G=N) H_seGN=N. LetV=N be an S-quasinormally embedded subgroup of G=N such that V=NHN=N. ThenV ˆV\HNˆN(V\H). We now show thatUˆV\HisS-quasinormally embedded inG. Letpbe an arbitrary prime dividingjUjandP2Syl_p(U). ThenP2Syl_p(V) since (jNj;jHj)ˆ1.

Hence PN=N2Syl_p(V=N). Let W=N be an S-quasinormal subgroup of G=N such that PN=N2Syl_p(W=N). Then PN=NˆW_pN=N for some Sylow p-subgroupW_p of W. Hence PNˆW_pN. SinceP and W_p are all Sylowp-subgroups ofPN,Pˆ(W_p)ⁿfor somen2N. By Lemma 2.1(2),W isS-quasinormal inG. Then, clearly,WⁿisS-quasinormal inG. Now since Pˆ(Wp)ⁿ 2Syl_p(Wⁿ), we see thatUisS-quasinormally embedded inG.

ThereforeUH_seG. It follows thatV=NˆUN=NH_seGN=Nand con- sequently (HN=N)_se(G=N) H_seGN=N.

(5) This is evident.

LEMMA 2.7. Let H be an SE-supplemented subgroup of G and N a normal subgroup of G.

(1) If HKG, then H is SE-supplemented in K.

(2) If NH, then H=N is SE-supplemented in G=N.

(3) If(jNj;jHj)ˆ1, then HN=N is SE-supplemented in G=N.

(4) H^xis SE-supplemented in G, for all x2G.

PROOF. LetTbe a subgroup ofGsuch thatHTˆGandH\THseG. (1) Since KˆH(T\K) and (T\K)\HˆT\HH_seGHseK by Lemma 2.6(2), (1) holds.

(2) Since (H=N)(NT=N)ˆG=H and (H=N)\(NT=N)ˆN(H\T)=N HseGN=N(H=N)se(G=N) by Lemma 2.6(3), H=N is SE-supplemented in G=N.

(3) Since (jNj;jHj)ˆ1,NT. Hence (T=N)\(HN=N)ˆN(T\H)=N NHseG=N(NH=N)se(G=N). This shows that HN=N is SE-supplemented inG=N.

(4) Since HTˆG, we have H^xT^xˆG. On the other hand, since H\TH_seG, H^x\T^xˆ(H\T)^x(H_seG)^xˆ(H^x)_seG by Lemma 2.6(5).

HenceH^xisSE-supplemented inG.

Recall that for a classFof groups, a groupGis said to be a minimal non- F-group ifG62 F but every proper subgroup ofGbelongs toF.

The following result about minimal non-F-groups is useful in our proofs.

LEMMA2.8 [28, VI, Theorem 25.4]. LetFbe a saturated formation and G a minimal non-F-group such that G^F is soluble. Then:

(a) G^F is a p-group for some prime p and G^F is of exponent p or ex- ponent4(if P is a non-abelian2-group).

(b) G^F=F(G^F)is a chief factor of G, and G^F=F(G^F)is a non-F-central in G.

LEMMA 2.9 [12, Theorem 2.4]. Let P be a group and a a p⁰-auto- morphism of P.

(1) If[a;V2(P)]ˆ1, thenaˆ1.

(2) If[a;V1(P)]ˆ1and either p is odd or P is abelian, thenaˆ1.

We use A(p 1) to denote the formation of all abelian groups of ex- ponent dividing p 1. The symbol Z_U(G) denotes the largest normal subgroup of G such that every chief factor of G below ZU(G) is cyclic (ZU(G)ˆ1 ifGhas no such non-identity normal subgroups).

LEMMA 2.10 [31, Lemma 2.2]. Let Ebe a normal p-subgroup of G. If EZ_U(G), then

(G=C_G(E))^{A(p 1)}Op(G=C_G(E)):

We useV(P) to denote the subgroupV₁(P) whenPis not a non-abelian 2-group; otherwise, letV(P)ˆV₂(P).

The following lemma may be proved based on some results in [23] on f-hypercentral action (see [28, Chapter II] or [11, Chapter IV, Section 6]).

For reader's convenience, we give a direct proof.

LEMMA 2.11. Let P be a normal p-subgroup of G. If either P=F(P)ZU(G=F(P))orVZU(G), then PZU(G).

PROOF. LetCˆC_G(P) andH=Kbe any chief factor ofGbelowP. Then O_p(G=C_G(H=K))ˆ1 by [40, Appendix C, Corollary 6.4].

Suppose that P=F(P)Z_U(G=F(P)). Then by Lemma 2.10, (G=CG(P=F(P)))^{A(p 1)} is a p-group. Hence (G=C)^{A(p 1)} is a p-group by Theorem 1.4 in [13, Chapter 5]. This implies thatG=C_G(H=K)2 A(p 1) and so jH=Kj ˆpby [40, Chapter 1, Theorem 1.4]. ThereforePZU(G).

Now assume that VZU(G). Then (G=CG(V))^{A(p 1)} is a p-group by Lemma 2.10. Hence (G=C)^{A(p 1)} is ap-group by Lemma 2.9. Thus we also havePZU(G).

LEMMA 2.12 [2, Lemma 4]. Let P be a p-subgroup of G, where p>2.

Suppose that all subgroups of P of order p are S-quasinormal in G. If a is a p⁰-element of N_G(P)nC_G(P), then a induces in P a fixed-point-free auto- morphism.

LEMMA2.13 [17, Theorem 3.1]. Let A, B, Ebe normal subgroups of G.

Suppose that GˆAB. If EZ_U(A)\Z_U(B)and(jG:Aj;jG:Bj)ˆ1, then EZ_U(G).

LEMMA2.14 [39]. Let G be a group and AG.

(1) If A is subnormal in G and A is ap-subgroup of G, then AOp(G).

(2) If A is subnormal in G and A is nilpotent, then AF(G).

The following lemma is well known.

LEMMA2.15. Let A;BG and GˆAB.

(1) GpˆApBpfor some Sylow p-subgroups Gp, Apand Bpof G, A and B, respectively.

(2) GˆAB^xfor all x2G.

LEMMA 2.16 [28, Chapter I, Lemma 4.1]. Let Q be an irreducible automorphism group of an elementary abelian p-group P of order pⁿ. Then Q is cyclic withjQj jpⁿ 1and n is the smallest positive integer such thatjQjdivides pⁿ 1.

Following Doerk and Hawkes [11], we useC^p(G) to denote the inter- section of the centralizers of all abelianp-chief factors ofG(C^p(G)ˆGifG has no such chief factors).

For every functionf of the form

f :P[ f0g ! fgroup formationsg;

()

we put, following [30], CLF(f)ˆ fGis a groupjG=G_S2f(0) and G=C^p(G)2f(p) for any prime p2p(Com(G))g. Here GS denotes the S- radical ofG (that is, the largest normal soluble subgroup ofG); Com(G) denotes the class of all abelian groups A such that A'H=K for some composition factorH=KofG.

It is well known that a formationFis a Bear-local formation if and only if there exists a functionf of the form (*) such thatF ˆCLF(f) (see, for example, [30, Theorem 1]).

LEMMA 2.17. Let F be a Baer-local formation containing all super- soluble groups and Ea subgroup of G such that G=E2 F. If Eis cyclic, then G2 F.

PROOF. Without loss of generality, we may assume that E is a minimal normal subgroup of G. ThenjEj ˆpfor some primep. By [30, Theorem 1], F ˆCLF(f) for some function of the form (*). It is clear that the group HˆE^j(G=CG(E)) is supersoluble. Hence H2 F and therebyG=C_G(E)2f(p). LetC^p=EˆC^p(G=E). ThenC^p(G)ˆC^p\C_G(E) by [11, Chapter A, Theorem 3.2]. But since G=E2 F, G=C^p' (G=E)=C^p(G=E)2f(p) and consequentlyG2 F.

3. Proof of Theorem 1.2

Theorem 1.2 is a special case of the following theorem whenp_iˆp(E), the set of all prime divisors ofjEj.

THEOREM3.1. Let Ebe a normal subgroup of G, p15p25. . .5pnthe set of all prime divisors of jEjandp_iˆ fp1;p2;. . .;p_ig. Suppose that for each p2p_iand for any Sylow p-subgroup P of E, the cyclic subgroups of P of prime order or order4are SE-supplemented in G. Then E has a normal Hallp⁰_i-subgroup Ep⁰_iand each chief factor of G between Eand Ep⁰_iis cyclic.

PROOF. Suppose that this theorem is false and consider a counter- example (G;E) for whichjGj ‡ jEjis minimal. Letpˆp₁be the smallest prime dividing jEj and P a Sylow p-subgroup of E. Let ZˆZ_U(G) and CˆCG(P). We proceed via the following steps.

(1) Eis p-nilpotent.

Without loss of generality, we may assume thatiˆ1.

IfE6ˆG, then the hypothesis is true for (E;E) by Lemma 2.7(1). Hence Eisp-nilpotent by the choice of (G;E). Now assume thatEˆGandGis notp-nilpotent. ThenGhas ap-closed Schmidt subgroupHˆHp^jHq[20, Chapter IV, Theorem 5.4]. We may assume thatHpP. By Lemma 2.8, Hp=F(Hp) is a non-central chief factor ofHandHpis a group of exponentp

or exponent 4 (ifpˆ2 andHpis non-abelian). HencejHp=F(Hp)j>psince pis the smallest prime dividingjHj.

LetFˆF(Hp),X=Fbe a minimal subgroup ofHp=F,x2X1Fand Lˆ hxi. Then jLj ˆp or jLj ˆ4. Hence L is SE-supplemented in G.

Suppose that L_seG6ˆL. Then for some proper subgroup Tof Gwe have LTˆG. Hence HˆL(H\T) and H\T6ˆH. Because FF(H), we have (H\T)F5H. Since the maximal subgroup ofL is contained in F, jH:(H\T)Fj ˆp. Hence jHp=Fj ˆ jH:(H\T)Fj ˆp. This contra- diction shows thatL_seGˆLisS-quasinormally embedded inGsinceLis cyclic. HenceLisS-quasinormally embedded inHby Lemma 2.4(1). Then by Lemma 2.5,LisS-quasinormal inH. It follows from Lemma 2.1 that LF=FˆX=F is S-quasinormal in H=F. This shows that every minimal subgroup of Hp=F is S-quasinormal in H=F and hence jHp=Fj ˆp by Lemma 2.11 in [29], a contradiction. HenceEisp-nilpotent.

(2) Let Ep⁰be a Hall p⁰-subgroup of E. Then Ep⁰is normal in G and the hypothesis holds for(G;E_p⁰)and for(G=E_p⁰;E=E_p⁰).

By (1),Ep⁰is characteristic inE. HenceEp⁰is normal inG. Clearly, the hypothesis holds for (G;Ep⁰). By Lemma 2.7(3), the hypothesis also holds for (G=Ep⁰;E=Ep⁰).

(3) EˆP is not a minimal normal subgroup of G.

Suppose thatE6ˆP. ThenE_p⁰ 6ˆ1. Hence every chief factor ofG=E_p⁰ below E=Ep⁰ is cyclic by the choice of (G;E). On the other hand, the minimality of (G;E) implies thatEp⁰has a normal Hallp⁰_i-subgroupV and each chief factor ofGbetweenEp⁰andVis cyclic. HenceVis a normal Hall p⁰_i-subgroup of E and each chief factor of G between E and V is cyclic, which contradicts the choice of (G;E):HenceEˆP. Suppose that Pis a minimal normal subgroup ofG. Then every minimal subgroupLofPisS- quasinormal inG. Indeed, sinceLisSE-supplemented inG, there exists some subgroupTofGsuch thatLTˆGandL\TL_seG. Suppose that L\Tˆ1. ThenT\Pis normal inGandjP:(T\P)j ˆp. It follows that T\Pˆ1 and sojEj ˆ jPj ˆp. Consequently, EZ. This contradiction shows that LT. Hence LˆL_seG is S-quasinormally embedded in G.

Then by Lemma 2.5, we see that every minimal subgroup ofP isS-qua- sinormal inG. ThereforejPj ˆp by Lemma 2.11 in [29], a contradiction.

Hence (3) holds.

(4) G has a non-identity normal subgroup RP such that P=R is a non-cyclic chief factor of G, RZ and V R for any normal subgroup V 6ˆP of G contained in P.

LetP=Rbe a chief factor ofG. ThenR6ˆ1 by (3) and the hypothesis holds for (G;R). ThereforeRZand soP=Ris not cyclic by the choice of (G;P)ˆ(G;E). Now letV 6ˆPbe any normal subgroup ofGcontained in P. Then V Z. If V6R, then from the G-isomorphism P=Rˆ VR=R'V=V\R we have PZ, which contradicts the choice of (G;E)ˆ(G;P). HenceV R.

(5) PO^p(G).

Suppose thatP6O^p(G). Then from theG-isomorphismO^p(G)P=O^p(G)' P=O^p(G)\P, we see thatGhas a cyclic chief factor of the formP=V, where O^p(G)\PV, which contradicts (4).

(6) V(P)ˆP.

IfV(P)5P, then by (4), VZ. HencePZby Lemma 2.11, which contradicts the choice of (G;P).

(7) There is a prime q6ˆp such that q dividesjG:Cj.

LetGpbe a Sylowp-subgroup ofG. Suppose thatjG:Cj ˆpⁿ. Then any chief factor ofGpbelowPis a chief factor ofG, which implies thatPZ.

The final contradiction.

By (6),V(P)ˆP. LetV1;V2;. . .;Vtbe the set of all cyclic subgroups of Pof orderpand order 4 (if Pis a non-abelian 2-group). We may assume thatP=Rˆ(V1R=R)(V2R=R) (VtR=R) andV_iR=Ris a group of orderp for all iˆ1;2;. . .;t. Suppose that for some i we have V_iTˆG, where T6ˆG. Then PˆV_i(T\P), where T\P6ˆP. LetNˆN_G(T\P). It is clear that jP:T\Pjis eitherpor 4. Hence eitherNˆGorjG:Nj ˆ2 and NPˆG. In the former case, G has a cyclic chief factor P=T\Pˆ V_i(T\P)=T\P'V_i=V_i\T\P. In the second case,Ghas a cyclic chief factorP=P\N. But in view of (4), both these cases are impossible. Hence by Lemma 2.5, V₁;V₂;. . .;V_t are S-quasinormal subgroups of G. This shows that ifQis a every Sylow subgroupQofG, thenViQˆQVifor every itand soViis subnormal inViQby Lemma 2.1(3). Consequently,Viis normal in V_iQ. Suppose that pˆ2. Then V_iQ is nilpotent and so QC_G(V_i). Therefore O^p(G)C_G(P=R). This implies C_G(P=R)ˆG, which contradicts (4). Hence p>2. We claim thatO^p(G)6ˆG. Indeed, if O^p(G)ˆG, then V₁R=R is normal in G=R by Lemma 2.2 and so P=RˆV1R=Ris cyclic, a contradiction. Next we show thatO^q(G)6ˆGfor some prime q6ˆp. Assume thatO^q(G)ˆGfor all primesq6ˆp. Then for every chief factor H=K of G of order p we have C_G(H=K)ˆG. In par-

ticular,LZ(G) for every minimal normal subgroupLofGcontained in R. This implies thatCP(a)6ˆ1 for everya2G. Hence by (4) and Lemma 2.12, G=C is a p-group, which contradicts (4). Thus O^q(G)6ˆG for some prime q6ˆp. By the choice of G, we have PZ_U(O^p(G)) and PZ_U(O^q(G)):Thus, by (5) and Lemma 2.13, we obtain that PZ. The final contradiction completes the proof.

4. Proof of Theorem 1.3

Theorem 1.3 is a special case of the following theorem whenp_iˆp(E).

THEOREM4.1. Let Ebe a normal subgroup of G, p15p25. . .5pnthe set of all prime divisors ofjEjandp_iˆ fp₁;p₂;. . .;p_ig. Suppose that for each p2p_i, the maximal subgroups of any Sylow p-subgroup of Eare SE- supplemented in G. Then Ehas a normal Hallp⁰_i-subgroup E_p⁰_iand each chief factor of G between Eand Ep⁰_iis cyclic.

PROOF. Assume that this theorem is false and let (G;E) be a coun- terexample for which jGj ‡ jEj is minimal. Let pˆp₁ be the smallest prime dividing jEj andP a Sylow p-subgroup of E. Let ZˆZ_U(G). We proceed the proof via the following steps.

(1) Eis p-nilpotent.

We may consider, without loss of generality, thatiˆ1. Assume thatE is notp-nilpotent. Then:

(a) EˆG.

Indeed, ifE5G, thenjEj ‡ jEj5jGj ‡ jEj. Hence the hypothesis is true for (E;E) by Lemma 2.7(1). The choice of (G;E) implies that Eisp-nil- potent, a contradiction.

(b) Op⁰(G)ˆ1.

Let DˆOp⁰(G). By Lemma 2.7(3), the hypothesis is true for (G=D;ED=D). Hence, if D6ˆ1, thenG=Disp-nilpotent by the choice of (G;E). ThereforeGisp-nilpotent, a contradiction.

(c) If PV5G, then V is p-nilpotent.

In fact, by Lemma 2.7(1), the hypothesis holds for V. Hence V is p- nilpotent by the choice ofG.

(d) Op⁰(L)ˆ1for all S-quasinormal subgroups L of G.

By Lemma 2.1(3), L is subnormal inG. It follows that O_p⁰(L) is sub- normal inG. HenceOp⁰(L)Op⁰(G)ˆ1 by Lemma 2.14(1).

(e) If N is an abelian minimal normal subgroup of G, then G=N is p- nilpotent.

In view of (b),Nis ap-group and soNP. Thus the hypothesis is true for (G=N;E=N) by Lemma 2.7(2). The choice of (G;E) implies thatG=Nis p-nilpotent.

(f) G is p-soluble.

In view of (e), we need only to show that G has an abelian minimal normal subgroup. Suppose that this is false. Then pˆ2 by the Feit- Thompson odd theorem. By Lemmas 2.1(3) and 2.14(1), we see that every non-identity subgroup ofPis notS-quasinormal inG. Hence for every non- identity S-quasinormally embedded inG subgroup LV, where V is a maximal subgroup ofP, and for anyS-quasinormal subgroupWofGsuch that L2Syl₂(W) we haveL6ˆW. Moreover,W_G6ˆ1. Indeed, ifW_Gˆ1, thenWis nilpotent by Lemma 2.3(3). HenceO₂⁰(W)6ˆ1, which contradicts (d). Note also that for any minimal normal subgroup N of G we have NPˆG(otherwise,Nis 2-nilpotent by (c), a contradiction). It follows that N is the unique minimal normal subgroup ofG. ThereforeNW(since W_G6ˆ1) and consequentlyN\PˆN\L.

Now we show thatV_seG6ˆ1 for any maximal subgroupV ofP. In fact, suppose that V_seGˆ1 and let T be a subgroup of G such that VTˆG and V\TVseGˆ1. ThenTis a complement of V in G. This induces thatTis 2-nilpotent since the order of a Sylow 2-subgroup ofTis equal to 2. We may, therefore, assume that TˆN_G(H1) for some Hall 2⁰- subgroup H₁ of G. It is clear that H₁N. By [15], any two Hall 2⁰- subgroups of N are conjugate in N. By Frattini Argument, GˆNT.

Then Pˆ(P\N)(P\T^x) for some x2G by Lemma 2.15(1). Let T1ˆT^xˆN_G(H1x). It is clear thatP\T16ˆP. Hence we can choose a maximal subgroup V1 inP containing P\T1. By the hypothesis, there exists a subgroup T₂ such that GˆV₁T₂, where V₁\T₂(V₁)_seG. If (V₁)_seGˆ1, then as above, we have that T₂ is 2-nilpotent and we may assume that T₂ˆN_G(H₂) for some Hall 2⁰-subgroup H₂ of G. By [15]

again, we have ((H1)^x)^yˆH2 for some y2G. Therefore, GˆVTˆ VT1ˆV1T2ˆV1T1yˆV1T1by Lemma 2.15(2) andPˆV1(P\T1)ˆV1. This contradiction shows that (V1)_seG6ˆ1. Let Lbe any non-identity S- quasinormally embedded subgroup ofGcontained inV₁andW be anS-

quasinormal subgroup ofGsuch thatL2Syl₂(W). ThenL\NˆP\N, which implies Pˆ(P\N)(P\T1)ˆ(L\N)(P\T1)V1, a contra- diction.

Therefore for every maximal subgroupVofPwe have (V₁)_seG6ˆ1. But then from above, we know thatN\PV. HenceN\PF(P) and soN is 2-nilpotent by [20, Chapter IV, Theorem 4.7], a contradiction. Hence, (f) holds.

The final contradiction for(1).

LetNbe any minimal normal subgroup ofG. Then in view of (b) and (f),Nis ap-group and soG=Nisp-nilpotent by (e). This implies thatN is the unique minimal normal subgroup ofGandN6F(G). HenceGis a primitive group and thereby NˆC_G(N)ˆF(G) by [11, Chapter A, Theorem 17.2]. LetMbe a maximal subgroup ofGsuch thatGˆN^jM.

LetM_p2Syl_p(M) andV be a maximal subgroup ofPsuch thatM_pV.

ThenVM6ˆGand soVM^x6ˆGfor allx2Gby Lemma 2.15(2). SinceV isSE-supplemented in G, there is a subgroupTof Gsuch thatVTˆG andV\TV_seG. Suppose thatV_seGˆ1. ThenTis a complement ofV in G. It follows thatjTpj ˆp, whereTp2Syl_p(T). ThenTisp-nilpotent since pis the smallest prime dividingjGj. HenceT_p⁰/T, whereT_p⁰ is a Hallp⁰-subgroup ofT. SinceGisp-soluble, any two Hallp⁰-subgroups of G are conjugate. Therefore there is an element x2G such that Tp⁰M^x. IfTpM^x, thenTM^xandGˆVTˆVM^x, a contradiction.

Hence Tp6M^x. But G=N'M^xN_G(Tp⁰) (since G=N is p-nilpotent) and T_pN_G(T_p⁰). ThereforeGˆ hM^x;T_pi ˆN_G(T_p⁰), which contradicts (b). Hence V_seG6ˆ1. Let L6ˆ1 be an S-quasinormally embedded sub- group ofGsuch thatLVandWanS-quasinormal subgroup ofGsuch thatL2Syl_p(W). Suppose that LˆW. Then by Lemma 2.2,NL^Gˆ L^PTp0 ˆL^PV, a contradiction. HenceL6ˆW. Then in view of (b) and Lemma 2.1(3), we have W_G6ˆ1. This implies that NLV and so V ˆVNˆP. This final contradiction shows that (1) holds.

(2) EˆP.

See (3) in the proof of Theorem 1.3.

(3) If N is a minimal normal subgroup of G contained in P, then P=NZU(G=N), N is the only minimal normal subgroup of G contained in P andjNj>p.

Indeed, by Lemma 2.7(2), the hypothesis holds onG=Nfor any minimal normal subgroup N of Gcontained inP. Hence P=NZU(G=N) by the

choice of (G;E)ˆ(G;P). IfjNj ˆp,PZU(G), a contradiction. IfGhas two minimal normal subgroupsRandNcontained inP, thenNR=RP=R and from theG-isomorphismRN=N'Nwe havejNj ˆp, a contradiction.

Hence, (3) holds.

(4) F(P)6ˆ1.

Suppose thatF(P)ˆ1. ThenPis an elementary abelianp-group. LetN1

be any maximal subgroup ofN. We show thatN1isS-quasinormal inG. Let Bbe a complement ofNinPandV ˆN₁B. ThenVis a maximal subgroup of P. Hence V isSE-supplemented inG. LetTbe a subgroup ofGsuch that GˆTVandT\VV_seG. IfTˆG, thenV ˆV_seGisS-quasinormal inGby Lemma 2.5. HenceV\NˆVseG\NˆN1B\NˆN1(B\N)ˆN1isS- quasinormal in G by Lemma 2.3(2). Now assume that T6ˆG. Then 16ˆT\P5P. SinceGˆVTˆPTandPis abelian,T\Pis normal inG.

HenceNT\PTand consequentlyN₁N\T\VN\V_seGN.

Clearly,N6V. HenceN₁ˆN\V_seG. By Lemma 2.5 and since the sub- group generated by allS-quasinormal inGsubgroup ofV is alsoS-quasi- normal inG(cf. [29, Lemma 2.8(1)]), we see thatV_seGisS-quasinormal inG.

Thus by Lemma 2.3(2), N1 isS-quasinormal inG. This shows that every maximal subgroup ofNisS-quasinormal inG. Hence some maximal sub- group ofNis normal inGby Lemma 2.11 in [29]. This contradiction shows thatF(P)6ˆ1.

The final contradiction.

By (4),F(P)6ˆ1. LetNbe a minimal normal subgroup ofGcontained in F(P):Then the hypothesis is still true forG=N. HenceP=NZ_U(G=N) by the choice of (G;E). This means that P=F(P)Z_U(G=F(P)). Then by Lemma 2.11, we obtain thatPZ. This final contradiction completes the proof.

5. Final remarks

In Section 1, we have seen that a large number of known results follow from our results. Now we consider some further applications.

1. A groupGis said to bequasisupersoluble[18] if for every its non- cyclic chief factorH=Kand everyx2G,xinduces an inner automorphism on H=K. It is cleat that every supersoluble group is quasisupersoluble.

Moreover, in [18] it is proved that the class of all quasisupersoluble groups

is a Baer-local formation. Hence from Theorems 1.7 and 1.8 we also obtain the following results.

THEOREM5.1. Let G be a group with normal subgroups XEsuch that G=Eis quasisupersoluble. Suppose that the cyclic subgroups of X of prime order or order 4 are SE-supplemented in G. If either XˆEor XˆF(E), then G is quasisupersoluble.

THEOREM5.2. Let G be a group with normal subgroups XEsuch that G=Eis quasisupersoluble. Suppose that the maximal subgroups of every Sylow subgroup of X are SE-supplemented in G. If either XˆEor XˆF(E), then G is quasisupersoluble.

2. Recall that a subgroupHofGis said to beweakly S-permutable (S- supplemented) in G[29] if there are a subnormal subgroup (a subgroup, respectively)T Gand anS-quasinormal subgroupH_sGofGcontained inH such thatHTˆGandH\THsG.

The following example shows that in general the set of SE-supple- mented subgroups of a group is wider than the set of all itsS-supplemented subgroups and the set of all its S-quasinormally embedded subgroups.

Consequently, the set of SE-supplemented subgroups of a group is also wider than the set of all S-quasinormal subgroups, the set of weakly S- permutable subgroups, the set of allc-normal subgroups and the set of all c-supplemented subgroups since these subgroups are either S-supple- mented orS-quasinormally embedded inG.

EXAMPLE 5.3. Let Ly be the Lyons simple group. Then jLyj ˆ 2⁸3⁷5⁶711313767. Hence in view of [14] there is a groupDwith minimal normal subgroup N such that CD(N)ˆNO67(D), D=N'Ly andNF(D). LetQbe a group of order 17. LetGˆDoQˆK^jQ, where Kis the base group of the regular wreath productG. ThenPˆF(K)ˆN^\ (we use here the terminology in [11, Chapter A]). Moreover, in view of [11, Chapter A, Proposition 18.5],Pis the only minimal normal subgroup ofG.

It is clear also thatjPj>67².

SincePis an elementary abelian 67-group, then in view of Maschke's theorem,PˆP₁P₂. . .P_t, whereP_iis a minimal normal subgroup ofPQfor alliˆ1;2;. . .;t. Suppose thatQCG(Pi) for alliˆ1;2;. . .;t.

ThenQC_G(P). HencePQˆPQˆC_G(P) is normal inGand soQis normal inG. This contradiction shows that for someiwe haveC_Q(P_i)ˆ1.

HenceS:ˆPirtimesQˆQ^S. SinceQis a Sylow 17-subgroup ofG, it isS- quasinormally embedded inG. HenceSˆS_seGand consequentlySisSE- supplemented in G. Suppose that P_i is an S-quasinormally embedded subgroup of G and letV be an S-quasinormal subgroup of G such that P_i2Syl_p(V). Since 17 divides 67‡1 and does not divide 67 1,jP_ij ˆ67² by Lemma 2.16. By Lemma 2.1(3),V is subnormal inGandV=V_Gis nil- potent by Lemma 2.3(2). Suppose that V_Gˆ1. Then V is a subnormal nilpotent subgroup ofG. It follows from Lemma 2.14(2) thatVP. Thus V ˆP_iisS-quasinormal inG. It is clear thatO⁶⁷(G)ˆG. By Lemma 2.2, we see thatP_iis normal inG. This induces thatPˆP_iand sojPj ˆ67², a contradiction. Therefore V_G6ˆ1 and soPV_G. But then PP_i, a con- tradiction again. ThusPiis not anS-quasinormally embedded subgroup of G. Similarly one can proved that any maximal subgroup ofP_iis not anS- quasinormally embedded subgroup of G. Hence Sis notS-quasinormally embedded in G and ifL is any non-identity S-quasinormally embedded subgroup of G contained in S, then LˆQ^x for some x2S. Moreover, obviously, S has no non-identity S-quasinormal in G subgroups, that is, S_sGˆ1. Now we show thatSis notS-complemented inG. Indeed, ifSisS- complemented inG, thenShas a complementTinGsinceS_sGˆ1. Clearly, TK. Hence KˆK\TSˆT(K\S)ˆTP_i. But since P_iPˆF(K), we obtainTˆK, which impliesT\S6ˆ1. This contradiction shows thatS is notS-supplemented inG.

Base on the above, we also see that the results in [29] in the case where the subgroupDin [29, Theorems] is of prime order or 4 can be obtained by our results in this paper.

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Manoscritto pervenuto in redazione il 26 Gennaio 2012.