Finite groups with weakly <span class="mathjax-formula">$s$</span> -semipermutable subgroups

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Finite Groups with Weakly s-Semipermutable Subgroups

CHANGWENLI

ABSTRACT- SupposeGis a finite groupandHis a subgroupofG.His said to bes- semipermutable inGifHGpGpH for any Sylowp-subgroupGp of Gwith (p;jHj)1;His called weaklys-semipermutable inGif there is a subgroupTof Gsuch thatGHT andH\Tiss-semipermutable inG. We investigate the influence of weakly s-semipermutable subgroups on the structure of finite groups. Some recent results are generalized and unified.

1. Introduction.

All groups considered in this paper will be finite. We use conventional notions and notation, as in Huppert [1].Gdenotes always a group,jGjis the order ofG,p(G) denotes the set of all primes dividingjGjandG_p is a Sylowp-subgroupofGfor somep2p(G). LetF be a class of groups. We callFa formation provided that (i) ifG2 FandH/G, thenG=H2 F, and (ii) if G=M and G=N are in F, then G=(M\N) is in F for any normal subgroupsM;NofG. A formationFis said to be saturated ifG=F(G)2 F implies thatG2 F. In this paper,U,N will denote the class of all supersolvable groups and the class of all nilpotent groups, respectively. As well- known results,U,N are saturated formations.

Two subgroupsHandKofGare said to be permutable ifHKKH.

A subgroup H of G is said to be s-permutable (or s-quasinormal, p- quasinormal) in G if Hpermutes with every Sylow subgroup of G [2].

Asaad, Ramadan and Shaalan proved in [3]: Suppose G is a solvable groupwith a normal subgroupHsuch that G=His supersolvable. If all

(*) Indirizzo dell'A.: School of Mathematical Science, Xuzhou Normal University, Xuzhou, 221116, China.

E-mail: lcwxz@xznu.edu.cn

The project is supported by the Natural Science Foundation of China (No:11071229) and the Natural Science Foundation of the Jiangsu Higher Educa- tion Institutions (No:10KJD110004).

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maximal subgroups of any Sylow subgroup ofF(H) ares-permutable in G, thenGis supersolvable. Later Asaad in [4] extended the result using formation theory: LetFbe a saturated formation containingU. Suppose thatGis a solvable groupwith a normal subgroupHsuch thatG=H2 F.

If all maximal subgroups of all Sylow subgroups of F(H) are s-permutable in G, then G2 F. As a generalization of s-permutable subgroup, the concept ofs-semipermutable subgroup is introduced. A subgroupH ofGis said to bes-semipermutable inGifHGpGpHfor any Sylowp- subgroupG_p of Gwith (p;jHj)1. Q. Zhang and L. Wang [5] obtained the following: LetFbe a saturated formation containingU. Sup p ose that Gis a groupwith a solvable normal subgroupHsuch thatG=H2 F. If all maximal subgroups of all Sylow subgroups ofF(H) ares-semipermutable in G, then G2 F. In recent years, it has been of interest to use sup- plementation properties of subgroups to characterize properties of a group. For example, Y. Wang [6] introduced the concept of c-supplemented subgroups and obtained the similar result in [7]: Let F be a saturated formation containing U. Sup p ose that G is a groupwith a solvable normal subgroup H such that G=H2 F. If all maximal subgroups of all Sylow subgroups of F(H) are c-supplemented in G, then G2 F.

There is no obvious general relationshipbetween s-semipermutable subgroupandc-supplemented subgroup. Hence it is meaningful to unify and generalize the two concepts and relate results. Recall that H is c- supplemented inGif there exists a subgroupK1 such thatGHK1 and H\K₁H_G, whereH_Gis the maximal normal subgroupofGcontained in H. In this case, writingKH_GK₁we haveGHK andH\KH_G; of course,H\Kiss-semipermutable inG. On the basis of this observation, we introduce a new embedding property:

DEFINITION1.1. A subgroup H of a group G is called weakly s-semipermutable in G if there is a subgroup T of G such that GHT and H\T is s-semipermutable in G.

In the present paper, we study the influence of weakly s-semipermutable subgroups on the structure of some groups. In particular, we give some new characterizations of supersolvability andp-nilpotency of a group(and, more general, a groupbelonging to a given formation of finite groups) by using the weaklys-semipermutability of some primary subgroups. As application, we unify and generalize a series of known results.

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2. Preliminaries.

LEMMA 2.1. Suppose that H is an s-semipermutable subgroup of a group G and N is a normal subgroup of G. Then

(a) H is s-semipermutable in K whenever HKG.

(b) If H is p-group for some prime p2p(G), then HN=N is s-semipermutable in G=N.

(c) If HOp(G), then H is s-permutable in G.

PROOF. (a) is [5, Property 1], (b) is [5, Property 2], and (c) is [5, Lemma 3].

LEMMA2.2. Let U be a weakly s-semipermutable subgroup of a group G and N a normal subgroup of G. Then

(a) If UHG, then H is weakly s-semipermutable in H.

(b) Suppose that U is a p-group for some prime p. If NU, then U=N is weakly s-semipermutable in G=N.

(c) Suppose U is a p-group for some prime p and N is a p⁰-subgroup, then UN=N is weakly s-semipermutable in G=N.

PROOF. By the hypotheses, there is a subgroup K of G such that GUKandU\Kiss-semipermutable inG.

(a) HH\UKU(H\K) andU\(H\K)U\K iss-semipermutable inHby Lemma 2.1(a). HenceUis weaklys-semipermutable inH.

(b) G=NUK=NU=NNK=Nand (U=N)\(KN=N)(U\KN)=N (U\K)N=Niss-semipermutable inG=Hby Lemma 2.1(b). HenceU=Nis weaklys-semipermutable inG=N.

(c) Since (jG:Kj;jNj)1, NK. It is easy to see that G=N UN=NKN=NUN=NK=N and (UN=N)\(K=N)(UN\K)=N (U\K)N=Niss-semipermutable inG=Nby Lemma 2.1(b). HenceUN=Nis weaklys-semipermutable inG=N.

LEMMA2.3([8], Lemma 2.6). Let H be a solvable normal subgroup of a group G(H61). If every minimal normal subgroup of G which is contained in H is not contained inF(G), then the Fitting subgroup F(H)of H is the direct product of minimal normal subgroups of G which are contained in H.

LEMMA2.4([7], Lemma 2.8). Let M be a maximal subgroup of G, P a normal p-subgroup of G such that GPM, where p a prime. Then P\M is a normal subgroup of G.

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LEMMA 2.5 ([9], Lemma 2.16). Let F be a saturated formation con- tainingU. Suppose that G is a group with a normal subgroup N such that G=N2 F. If N is cyclic, then G2 F.

LEMMA 2.6([9], Lemma 2.20). Let A be a p⁰-automorphisms of a p- group P, where p is an odd prime. Assume that every subgroup of P with prime order is A-invariant. Then A is cyclic.

LEMMA2.7([1], III, 5.2 and IV, 5.4). Suppose G is a group which is not nilpotent but whose proper subgroups are all nilpotent. Then

(a) G has a normal Sylow p-subgroup P for some prime p and GPQ, where Q is a non-normal cyclic q-subgroup for some prime q6p.

(b) P=F(P)is a minimal normal subgroup of G=F(P).

(c) If P is non-abelian and p>2, then the exponent of P is p; If P is non-abelian and p2, then the exponent of P is4.

(d) If P is abelian, then the exponent of P is p.

(e) F(P)Z(P).

LEMMA2.8([8], Lemma 3.12). Let P be a Sylow p-subgroup of a group G, where p is the smallest prime dividingjGj. If G is A₄-free andjPj4p², then G is p-nilpotent.

3. Results.

THEOREM3.1. LetF be a saturated formation containingU. A group G2 Fif and only if there is a normal subgroup E of G such that G=E2 F and every cyclic subgrouphxiof any noncyclic Sylow subgroup of E with prime order or order4( if the Sylow2-subgroup is non-abelian ) is weakly s-semipermutable in G.

PROOF. We need only to prove the sufficiency part since the necessity part is evident. Suppose that the assertion is false and let (G;E) be a counterexample for whichjGjjEjis minimal. Then

(1)Eis solvable.

LetKbe any proper subgroup ofE. ThenjKj5jGjandK=K2 U. Let hxi be any cyclic subgroupof any noncyclic Sylow subgroupof K with prime order or order 4 ( if the Sylow 2- subgroup is non-abelian ). It is clear thathxiis also a cyclic subgroupof a noncyclic Sylow subgroupofEwith prime order or order 4. By the hypothesis,hxiis weaklys-semipermutable

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inG. By Lemma 2.2,hxiis weaklys-semipermutable inK. This shows that the hypothesis still holds for (U;K). By the choice ofG,Kis supersolvable.

By [10, Theorem 3.11.9],Eis solvable.

(2)G^F is ap-group,whereG^F is theF-residual ofG.G^F=F(G^F) is a chief factor ofGand exp(G^F)por exp(G^F)4 (ifp2 andG^Fis non-abelian).

SinceG=E2 F,G^F E. LetMbe a maximal subgroupofGsuch that G^F 6M (that is, M is an F-abnormal maximal subgroupof G). Then GME. We claim that the hypothesis holds for (F;M). In fact, M=M\EME=EG=E2 Fand by the similar argument as above, we can prove that the hypothesis holds for (F;M). By the choice ofG,M2 F.

Thus (2) holds by [10, Theorem 3.4.2].

(3)hxiiss-permutable inGfor any elementx2G^F.

Letx2G^F. Then the order ofxispor 4 by step(2). By the hypothesis, hxiis weaklys-semipermutable inG. Then there is a groupTofGsuch that G hxiTandhxi \Tiss-semipermutable in G. It follows thatG hxiT andG^F G^F\GG^F \ hxiT hxi(G^F \T). SinceG^F=F(G^F) is abelian, (G^F \T)F(G^F)=F(G^F)/G=F(G^F). SinceG^F=F(G^F) is a chief factor ofG, G^F \TF(G^F) or G^F (G^F \T)F(G^F)G^F \T. If G^F \TF(G^F), hxi G^F/G. In this case, hxi is s-permutable in G. Now assume that G^F G^F \T. Then TG and hxi hxi \T is s-semipermutable in G.

Sincehxi G^F O_p(G),hxiiss-permutable inGby Lemma 2.1.

(4)jG^F=F(G^F)j p.

Assume thatjG^F=F(G^F)j 6pand letL=F(G^F) be any cyclic subgroup of G^F=F(G^F). Let x2LnF(G^F). ThenL hxiF(G^F). Since hxi iss-permutable inGby step(3),L=F(G^F) iss-permutable inG=F(G^F). It follows from [9, Lemma 2.11] that G^F=F(G^F) has a maximal subgroupwhich is normal inG=F(G^F). But this is impossible sinceG^F=F(G^F) is a chief factor ofG. ThusjG^F=F(G^F)j p.

(5) The final contradiction.

Since (G=F(G^F))=(G^F=F(G^F))G=G^F 2 F,G=F(G^F)2 F by Lemma 2.5. AsF(G^F)F(G) andFis a saturated formation, we haveG2 F. The final contradiction completes the proof.

COROLLARY 3.2 [14], Theorem 3.4). Let F be a saturated formation containing U, the class of all supersolvable groups. If every cyclic subgroup ofG^F with prime order or order4isc-normal inG, thenG2 F.

COROLLARY 3.3 ([15], Theorem 4.2). If every cyclic subgroup of a group G with prime order or order 4 is c-normal in G, then G is supersolvable.

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COROLLARY 3.4 ([16], Theorem 4.1). If every cyclic subgroup of G^U with prime order or order 4 is c-supplemented in G, then G is supersolvable.

COROLLARY 3.5 ([17], Theorem 1). Let F be a saturated formation containingU, the class of all supersolvable groups. If there is a normal subgroupHofGsuch thatG=H 2 Fand every cyclic subgroup ofHwith prime order or order4iss-permutable inG, thenG2 F.

COROLLARY3.6 ([21], Theorem 3.9). LetF be a saturated formation containing U, the class of all supersolvable groups. Then G2 F if and only if there is a normal subgroup H ofGsuch that G=H2 F and the subgroups prime order or order4ofH with arec-normal inG.

COROLLARY3.7 ([19], Theorem 3.1). LetGbe a group andNa normal subgroup of a groupGsuch thatG=N is supersolvable. If every minimal subgroup ofEisc-supplemented inGand if every cyclic subgroup of order 4ofN isc-normal inG, thenGis supersolvable.

THEOREM3.8. LetF be a saturated formation containingU. A group G2 Fif and only if there is a solvable normal subgroup H of G such that G=H2 F and every cyclic subgrouphxiof any noncyclic Sylow subgroup of F(H)with prime order or order4(if the Sylow2-subgroup is non-abelian) is weakly s-semipermutable in G.

PROOF. It is clear that the condition is necessary. We only need to prove that it is sufficient. Suppose that the assertion is false and let (G;H) be a counterexample for which jGjjHj is minimal. Let p be the smallest prime divisor ofjF(H)jandPthe Sylowp-subgroupofF(H). ThenP/G.

Now we proceed with our proof as follows:

(1) F(H)6HandC_H(F(H))F(H).

IfF(H)H, thenG2 F by Theorem 3.1, a contradiction. Obviously, C_H(F(H))F(H) sinceHis solvable.

(2) Let V=PF(H=P) and Q be a Sylow q-subgroupof V, where qkV=Pj. Thenq6pand eitherQF(H) orp>qandC_Q(P)1.

SinceV=Pis nilpotent,QP=PcharV=Pand soQP/H. Then, it is easy to see that p6q. By Theorem 3.1, PQ is supersolvable. If q>p, then Q/PQand soQF(H). Now assume thatp>q. Thenp>2. Sincepis the minimal prime divisor ofjF(H)j,F(H) is aq⁰-group. LetRbe a Sylowr-

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subgroupofF(H) wherer6p. Thenr6qand so [R;Q]P. Assume that for some x2Q , we havex2CH(P). Since V=P is nilpotent, [R;hxi] [R;hxi;hxi]1 by [11, Chapter 5, Theorem 3.6]. Hencex2C_H(F(H)). By (1),C_H(F(H))F(H) and soC_Q(P)1.

(3) p>2. If p2, then by (2), we see that F(H=P)F(H)=P and 2j⁴⁴jF(H=P)j. This implies that ifhxiP=Pis an arbitrary minimal subgroup ofF(H)=P, thenjxj r, wherer62. By Lemma 2.2, every minimal sub- groupofF(H=P) is weaklys-semipermutable inG=P. Hence (G=P;H=P) satisfies the hypothesis. The minimal choice of (G;H) implies that G=P2 F. Hence by Theorem 3.1,G2 F, a contradiction. Thus, (3) holds.

(4) Final contradiction. Let V=PF(H=P) andQ be a Sylow q-subgroupof V, where qkV=Pj. Then by (2), either QF(H) or p>q and C_Q(P)1. In the second case,Q is cyclic by (3) and Lemma 2.6. Hence every Sylow subgroupof F(H=P) either is cyclic or is contained inF(H).

Moreover by (2),pj⁴⁴jF(H=P)j. LetK=Pbe a cyclic subgroupof a non-cyclic Sylow subgroupofF(H=P) with prime order. Then it is easy to see that K=P hxiP=P, where hxiis a cyclic subgroupof some non-cyclic Sylow subgroupofF(H) with prime order. By hypothesis,hxiis weaklys-semipermutable in G. Hence hxiP=P is weakly s-semipermutable in G=P by Lemma 2.2. This shows that (G=P;H=P) satisfies the hypothesis. The minimal choice of (G;H) implies that G=P2 F. Therefore, G2 F by Theorem 3.1. The final contradiction completes the proof.

COROLLARY3.9 ([22], Theorem 3). LetGbe a group andEa solvable normal subgroup of G such thatG=E is supersolvable. If all minimal subgroups and all cyclic subgroups with order4ofF(E)arec-normal in G, thenGis supersolvable.

COROLLARY 3.10 ([23], Theorem 2). Let F be a saturated formation containingU. Suppose thatGis a group with a solvable normal subgroup Hsuch thatG=H 2 F. If all minimal subgroups and all cyclic subgroups with order4ofF(H)isc-normal inG, thenG2 F.

COROLLARY 3.11 ([24], Theorem 3). Let F be a saturated formation containing U. A groupG2 F if and only if there is a solvable normal subgroupHofGsuch thatG=H 2 Fand the subgroups of prime order or order4ofF(H)isc-normal inG.

COROLLARY3.12 ([7], Theorem 4.1). LetF be a saturated formation containingU. Suppose thatGis a group with a solvable normal subgroup

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Hsuch thatG=H2 F. If all minimal subgroups and all cyclic subgroups with order4ofF(H)isc-supplemented inG, thenG2 F.

COROLLARY 3.13 ([5], Theorem 4). Let F be a saturated formation containingU. Suppose thatGis a group with a solvable normal subgroup Hsuch thatG=H2 F. If all minimal subgroups and all cyclic subgroups with order4ofF(H)iss-semipermutable inG, thenG2 F.

COROLLARY 3.14 ([27], Corollary 1). Suppose G is a solvable group with a normal subgroup H such that G=H is supersolvable. If every subgroup ofF(H)of prime order or order4is s-permutable in G, thenGis supersolvable.

COROLLARY3.15 ([27], Theorem 1). A groupG2 Fif and only if there is a solvable normal subgroup H of G such that G=H2 F and the subgroups of prime order or order4ofF(H)iss-permutable inG.

THEOREM3.16. Suppose G is a group. If every subgroup of G with prime order is contained in Z₁(G)and every cyclic subgroup of G with order4is weakly s-semipermutable in G or lies in Z1(G), then G is nilpotent.

PROOF. Suppose that the theorem is false, and let G be a counterexample of minimal order. Let Hbe an arbitrary proper subgroup of G and hxi be a cyclic subgroupof H with prime order or order 4, then hxi Z₁(G)\HZ₁(H). By Lemma 2.2,hxiis weaklys-semipermutable in H. Thus H satisfies the hypotheses of the theorem in any case. The minimal choice ofGimplies thatHis nilpotent, thusGis a groupwhich is not nilpotent but whose proper subgroups are all nilpotent. By Lemmas 2.7, GPQ, whereP is normal inGfor some p2p(G) andQis non-normal cyclic. Then we have:

(1)p2 and every element with order 4 is weaklys-semipermutable inG.

Ifp>2, by Lemma 2.7, exp(P)p. ThusPZ1(G) by hypotheses.

Therefore,G=Z₁(G) is nilpotent. It follows thatGis nilpotent, a contradiction. If every element with order 4 ofGlies inZ₁(G), thenPZ₁(G), we have the same contradiction. Thus (1) holds.

(2) For everyx2PnF(P), we have(x)4.

If not, there existsx2PnF(P) and(x)2. DenoteM hxi^GP. Then MF(P)=F(P)/G=F(P), we have thatPMF(P)MZ(G) asP=F(P) is a minimal normal subgroupofG=F(P) by Lemma 2.7, a contradiction.

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(3) Final contradiction.

By (2), every elementxinPnF(P) is order 4. Thenxis weaklys-semipermutable inG. Thus there is a subgroupTofGsuch thatG hxiTand hxi \Tis semipermutable inG. HencePP\GP\ hxiT hxi(P\T).

Since P=F(P) is abelian, we have (P\T)F(P)=F(P)/G=F(P). Since P=F(P) is the minimal normal subgroupof G=F(P), P\TF(P) or P(P\T)F(P)P\T. If P\TF(P), then hxi P, a contraction with Lemma 2.7(d). Therefore PP\T. Then TG and so hxi is s- semipermutable in G. Sincehxi O_p(G)P, hxi is permutable inGby Lemma 2.1. We have hxiQ is a proper subgroup of G and so hxiQ hxi Q. Thereforex2N_G(Q), and it follows thatPN_G(Q) and GPQ, the final contradiction.

THEOREM3.17. LetF be a saturated formation such thatN F. Let G be a group such that every cyclic subgroup of G^F with order4is weakly s-semipermutable in G. Then G2 F if and only if every subgroup of G^F with prime order lies in theF-hypercenter ZF(G)of G.

PROOF. IfG2 F, thenZF(G)Gand we are done. So we only need to prove that the converse is true. Assume the converse is false and letGbe a counterexample of minimal order. ThenG2 F= . Letxbe an element with prime order ofG^F. Thenx2ZF(G)\G^F which is contained inZ(G^F) by [12, IV, 6.10]. By Lemma 2.2, every cyclic subgroupofG^F with order 4 is weaklys-semipermutable inG^F. Theorem 3.16 implies thatG^Fis nilpotent.

IfG^F F(G), thenG=F(G)2 F, henceG2 FsinceFis saturated. This is a contradiction. So there exists a maximal subgroupofG, sayM, such that GMG^F MF(G). By [13, Theorem 3.5], we may chooseMto be anF- critical maximal subgroup. Since G=M_G2 F= , it follows that ZF(G)M.

Moreover, aG-chief factorA=BbelowZF(G) is actually anM-chief factor and Aut_M(A=B) is isomorphic to Aut_G(A=B) becauseF(G) centralizesA=B.

Consequently Z_F(G) is contained in Z_F(M). By [12, IV, 1.17],M^F G^F. HenceMsatisfies the hypotheses of the theorem. The minimal choice ofG implies thatM2 F. By [10, Theorem 3.4.2],Ghas the following properties:

(a) G^F is ap-group, for some primep.

(b) G^F=F(G^F) is a minimal normal subgroupofG=F(G^F).

(c) IfG^F is abelian, thenG^F is an elementary abelianp-group.

(d) Ifp>2, then exp(G^F)p; ifp2, then exp(G^F)2 or 4.

If G^F is abelian, then G^F is an elementary abelian subgroupby (c).

Hence, by hypothesis, we have that G^F ZF(G). It follows that G2 F.

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This contradiction shows that G^F is nonabelian. If exp(G^F)p, then G^F ZF(G) by hypothesis and consequentlyG2 F, a contradiction again.

Thus,G^F is a non-abelian 2-groupand exp(G^F)4.

Let x be an arbitrary element of G^FnF(G^F). Then jxj 4. Indeed, suppose that there exists an elementx2G^FnF(G^F) such thatjxj 2. Let T hxi^G. ThenTG^F andTF(G^F)=F(G^F) is normal inG=F(G^F). Since G^F=F(G^F) is a chief factor ofG,G^F T, which contradicts the fact that exp(G^F)4. Now we will provehxiiss-permutable inG. By hypothesis, hxiis weaklys-semipermutable inG. Hence there exists a subgroupKofG such thatG hxiK andhxi \K iss-semipermutable inG. It follows that G^F G^F \GG^F \ hxiK hxi(G^F \K). Since G^F=F(G^F) is abelian, (G^F \K)F(G^F)=F(G^F)/G=F(G^F). SinceG^F=F(G^F) is a chief factor ofG, G^F \KF(G^F) orG^F (G^F \K)F(G^F)G^F\K. IfG^F \KF(G^F), hxi G^F /G. In this case, hxi is s-permutable in G. Now assume that G^F G^F \K. Then KG and hxi is s-semipermutable in G. Since hxi G^F O_p(G),hxiiss-permutable inGby Lemma 2.1.

Thus for anyq2p(G),q62,hxiis normalized by any Sylowq-subgroup QofM. SoQacts onhxiby conjugation. But the automorphism group of a cyclic groupof order 4 is a cyclic groupof order 2, soQacts trivially onhxi andQcentralizeshxi. Thushxiis centralized byO²(M), it implies thatG^Fis centralized byO²(M). HenceO²(M)/GasGMG^F. It follows thatG=M_G is a 2-group. Therefore, G=MG2 F since N F, a final contradiction.

This completes the proof of Theorem 3.17.

COROLLARY3.18 ([14], Theorem 3.2). LetF be a saturated formation such thatN F. LetGbe a group such that every cyclic subgroup ofG^F with order4isc-normal inG. ThenG2 Fif and only if every subgroup of G^F with prime order lies in theF-hypercenterZF(G)ofG.

COROLLARY3.19 ([18], Theorem 4.4). LetF be a saturated formation such thatN F. LetGbe a group such that every cyclic subgroup ofG^F with order 4 isc-supplemented in G. Then G2 F if and only if every subgroup ofG^F with prime order lies in theF-hypercenterZ_F(G)ofG.

COROLLARY3.20 ([19], Theorem 2.5). Suppose thatp is a prime and KG^N be the nilpotent residual of G. Then G isp-nilpotent if every minimal subgroup ofKis contained inZ₁(G)and every cyclichxiofK with order4isc-supplemented inG.

COROLLARY 3.21 ([20], Theorem 2.4). Let G be a finite group and KG^N be the nilpotent residual ofG. ThenGis nilpotent if and only if

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every minimal subgrouphxiofKlies in the hypercenterZ1(G)ofGand every cyclic element ofP with order4isc-normal inG.

THEOREM3.22. Let p be the smallest prime dividing the order of a group G and N a normal subgroup of G such that G=N is p-nilpotent. If G is A₄-free and every subgroup of N with order p² is weakly s-semipermutable in G, then G is p-nilpotent.

PROOF. Assume that the Theorem is false and let G be a counterexample of minimal order. Then:

(1) Every proper subgroup ofGisp-nilpotent.

By Lemma 2.8, we see thatjNj_p>p². LetLbe a proper subgroup ofG.

SinceL=(L\N)LN=NG=N,L=(L\N) isp-nilpotent. IfjL\Nj_p4p², thenLisp-nilpotent by Lemma 2.8. IfjL\Nj_p>p², then every subgroupof L\Nof orderp²is weaklys-semipermutable inLby Lemma 2.2. HenceLis p-nilpotent by the choice ofG. This shows thatGis a minimal non-p-nilpotent group.

(2) G has the following properties: (i) GPQ, where PG^N is a normal Sylowp-subgroupofGandQis a non-normal cyclic Sylowq-subgroupof G; (ii) P=F(P) is a minimal normal subgroupofG=F(P); (iii) If p>2, then the exponent ofPisp; ifp2, then the exponent ofPis 2 or 4;

(iv)F(P)Z(P); (v)p³dividing the order ofP; (vi)PN.

By Step(1) and [1, Theorem IV. 5.4], G is a minimal non-nilpotent group. Hence (i)-(iv) follow directly from Lemma 2.7. (v) follows from Lemma 2.8. (vi) is clear sincePG^N is thep-nilpotent residual ofGand G=Nisp-nilpotent.

(3) IfHis a subgroupofPof orderp², thenHiss-permutable inG.

LetHbe a subgroupofPof orderp². By the hypothesis,His weaklys- semipermutable inG. Then there is a subgroupTofGsuch thatGHTand H\T iss-semipermutable in G. HencePP\GP\HTH(P\T).

SinceP=F(P) is abelian, we have (P\T)F(P)=F(P)/G=F(P). By step(2) (ii), P\TF(P) or P(P\T)F(P)P\T. If P\TF(P), then HP/G. In this case,Hiss-permutable inG. IfPP\T, thenTG and soHiss-semipermutable inG. SinceHPOp(G),Hiss-permutable inGby Lemma 2.1.

(4) There exists a subgroup H of P such that jHj p² which is not contained inF(P).

IfF(P)1, then it is clear. Hence we may assume thatF(P)61. If jPj p³, then clearlyPhas a maximal subgroupof orderp². SincePis not cyclic by Burnside's Theorem [11, Theorem 4.3, P.252],Phas at least two

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different maximal subgroupsP1 andP2. IfP1 andP2are all contained in F(P), thenPP1P2F(P), a contradiction. Hence, we can assume that jPj>p³. Letx2PnF(P) anda2F(P) wherejaj p. SinceF(P)Z(P), hxihai G. By Step(2), we see thatjxj por 4. Ifjxj 4, we can choose H hxi. If jxj p, then jhxihaij4p². If jhxihaij p, then hxi hai, a contradiction. Hencejhxihaij p². Therefore (4) holds.

By Step(2),G[P]Q. By Step(4), there exists a subgroupHof P with orderp² such thatH6F(P). Then by (3),Hiss-permutable inG.

Hence HQQH. Then HH(Q\P)HQ\P/HQ. It follows that QN_G(H). On the other hand, since P=F(P) is abelian, HF(P)=F(P)/P=F(P). This implies thatHF(P)=F(P)/G=F(P). However, sinceP=F(P) is chief factor ofG, we obtain thatHF(P)P and conse- quentlyHP, a contradiction.

THEOREM 3.23. Let F be a saturated formation containing U, the class of all supersolvable groups. Suppose that G is a group with a solvable normal subgroup H such that G=H2 F. If all maximal subgroups of all Sylow subgroups of F(H)are weakly s-semipermutable in G, then G2 F.

PROOF. Assume that the assertion is false and let (G;H) be a counter example withjGjjHjis minimal. LetPbe an arbitrary Sylowp-subgroupof F(H). ClearlyP/G. We proceed the proof by the following steps.

(1)P\F(G)1.

If P\F(G)61, then P\F(G)R/G. Obviously, (G=R)=(H=R) G=H2 F and F(H=R)F(H)=R. Let P1=R be a maximal subgroupof the Sylowp-subgroupP=R. ThenP1is a maximal subgroupof the Sylow p-subgroupP. By hyp othesis,P1is weaklys-semipermutable inG. Hence P₁=Ris weaklys-semipermutable inG=Rby Lemma 2.2. LetM₁=Rbe a maximal subgroupof the Sylowq-subgroupofF(H)=R, wherep6q. It is clear that M1Q1R, where Q1 is a maximal subgroupof the Sylow q- subgroupof F(H). Then Q1 is weakly s-semipermutable in G by hypothesis. Hence M1=R is weakly s-semipermutable in G=R by Lemma 2.2. Now we have proved that (G=R;H=R) satisfies the hypotheses of the theorem. Therefore G=R2 F by minimal choice of (G;H). Since RF(G) and F is a saturated formation, we have that G2 F, a contradiction. Thus (1) holds.

(2)PR1R2 Rm, where R_i(i1;2; ;m) is some normal subgroupofGof orderp.

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Since P/G and P\F(G)1, PR1R2 Rm, where Ri(i1;2; ;mis an abelian minimal normal subgroupofGby Lemma 2.3. We now prove thatjR_ij p. SinceR_i6F(G), there exists a maximal subgroupMofGsuch thatGR_iMandR_i\M1. LetM_pbe a Sylowp- subgroupofMandG_pM_pR_i. ThenG_pis a Sylowp-subgroupofG. Let G₁ be a maximal subgroupof G_p containing M_p and P₁ G₁\P. Then jP:P1j jP:G1\Pj jPG1:G1j jGp:G1j pand soP1is a maximal subgroupof P. We also have that P1Mp(G1\P)MpG1\PMp G₁\G_pG₁ andP₁\M_pP\G₁\MP\M_p. By hypothesis,P₁ is weaklys-semipermutable inG. Hence there exists a subgroupTofGsuch that GP₁T and P₁\T is s-semipermutable in G. By Lemma 2.1(c), P1\Tiss-permutable inG. Then, for an arbitrary Sylowq-subgroupGqofG with q6p, (P1\T)GqGq(P1\T). Hence P1\T(P1\T)(P\Gq) P\(P₁\T)G_q/(P₁\T)G_q. It follows that G_qN_G(P₁\T). On the other hand,P\T/TandP\T/PsincePis abelian. HenceP\T/PTGand consequentlyP₁\TG₁\P\T/G₁. It follows thatP₁\T/G₁PG_p. This shows that bothGpandGqare contained inNG(P1\T). The arbitrary choice of q implies that P1\T/G and so P1\T(P1)_G. Assume that P1\T5(P1)_Gand letN(P1)_GT. ThenGP1TP1(P1)_GTP1Nand P₁\NP₁\(P₁)_GT(P₁)_G(P₁\T)(P₁)_G. This shows that there always exists a subgroupKofGsuch thatGP₁KandP₁\K(P₁)_G.

SincePis abelian,P1(P\M)/P. ThusP1(P\M)PorP1(P\M) P1. If P1(P\M)P, then GPMP1(P\M)MP1M and so PP\P1MP1(P\M)P1(P\G1\M)P1(P1\M)P1, a contradiction. Hence P₁(P\M)P₁ and so P\MP₁. Since P\M/G by Lemma 2.4, P\M(P₁)_GP₁\K.

Assume thatK5G. LetK1 be a maximal subgroupofGcontainingK.

ThenP\K1/Gby Lemma 2.4. Hence (P\K1)Mis a subgroupofG. Since M5G, (P\K1)MGor (P\K1)MM. If (P\K1)MGPM, then PP\(P\K₁)M(P\K₁)(P\M)P\K₁ since P\M(P₁)_G P₁\KP\K₁. It follows thatPK₁and henceGPK PK₁K₁, a contradiction. If (P\K1)MM, then P\K1M and so P1\K P\KP\K1P\K1\MP\MP1\K. Hence P1\K P\K.

SinceGPKP1K,jG:Pj jPK:Pj jK:(P\K)j jK:(P1\K)j jP₁K:P₁j jG:P₁j, which is impossible. Thus K G. It follows that P₁\KP₁(P₁)_G/G. Consequently,P₁\R_i/G. But sinceG_pR_iM_p R_iG₁andG₁is a maximal subgroupofG_pcontainingM_p, we haveR_i6P₁ G1\P. The minimal normality of Ri implies that P1\Ri1. Hence jR_ij jR_i:(P1\R_i)j jR_iP1:P1j jR_i(P\G1):P1j j(P\R_iG1):P1j jP\Gp:P₁j jP:P₁j p. ThereforeR_iis a cyclic groupof orderp.

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Let RiH and C0CH(Ri). We claim that the hypothesis holds for (G=R_i;C0=R_i). Indeed, since G=C_G(R_i) Aut(R_i) is abelian, G=C_G(R_i)2 F. Consequently, G=C₀G=(H\C_G(R_i))2 F. Besides, since R_iZ(C₀) and F(H)C₀, we have F(H)F(C₀). Thus F(C₀=R_i) F(H)=R_i. Let P=R_i be a Sylow p-subgroupof F(H)=R_i, where P is a Sylow p-subgroupof F(H) and G1=Ri is a maximal subgroupof P=R_i. Then P1 is a maximal subgroupof P. By hyp othesis, P₁ is weakly s-semipermutable in G. Hence P₁=R_i is weakly s- semipermutable in G=R_i by Lemma 2.2. Now assume that QR_i=R_i is the Sylow q-subgroupof F(H)=R_i, where q6p and Q is the Sylow q- subgroupof F(H). Then every maximal subgroupof QRi=Ri is of the form of Q1R_i=R_i, where Q1 is a maximal subgroupof Q. By hyp othesis and Lemma 2.2, we see that Q₁R_i=R_i is weakly s-semipermutable in G=R_i. This shows that (G=R_i;C₀=R_i) satisfies the condition of the theorem. The minimal choice of (G;H) implies that G2 F by Lemma 2.5. The final contradiction completes the proof.

COROLLARY 3.24 ([5], Theorem 2). Let F be a saturated formation containingU, the class of all supersolvable groups. Suppose that Gis a group with a solvable normal subgroup H such that G=H2 F. If all maximal subgroups of all Sylow subgroups ofF(H)ares-semipermutable inG, thenG2 F.

COROLLARY3.25 ([4], Theorem 1.4). LetF be a saturated formation containingU. Suppose thatGis a solvable group with a normal subgroup Hsuch thatG=H 2 F. If all maximal subgroups of all Sylow subgroups of F(H)ares-permutable inG, thenG2 F.

COROLLARY3.26 ([23], Theorem 1). Let F be a saturated formation containingU, the class of all supersolvable groups. Suppose that Gis a group with a solvable normal subgroup H such that G=H2 F. If all maximal subgroups of all Sylow subgroups ofF(H)arec-normal inG, thenG2 F.

COROLLARY3.27 ([7], Theorem 4.5). LetF be a saturated formation containingU, the class of all supersolvable groups. Suppose that Gis a group with a solvable normal subgroup H such that G=H2 F. If all maximal subgroups of all Sylow subgroups ofF(H)arec-supplemented in G, thenG2 F.

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COROLLARY3.28 ([25], Theorem 1.6). LetF be a saturated formation containingU, the class of all supersolvable groups. Suppose thatGis a group with a solvable normal subgroup H such that G=H2 F. If all maximal subgroups of all Sylow subgroups ofF(H)are complemented in G, thenG2 F.

COROLLARY3.29 ([22], Theorem 2). LetGbe a group andEa solvable normal subgroup of G such that G=E is supersolvable. If all maximal subgroups of the Sylow subgroups of F(E)arec-normal in G, then Gis supersolvable.

COROLLARY 3.30 ([28], Theorem 1.2). Suppose that G is a solvable group with a normal subgroup H such thatG=H is supersolvable. If all maximal subgroups of every Sylow subgroup ofF(H)are complement in G, thenGis supersolvable.

Acknowledgement. The author would like to thank the referee for his or her helpful comments and suggestions which have improved the original manuscript to its present form.

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Manoscritto pervenuto in redazione il 9 novembre 2010.