Lijun HUO Wenbin GUO Alexander A.MAKHNEV

1 Introduction

Throughout this paper,all groups considered are finite andGdenotes a finite group.Recall that a subgroupHofGis said to beS-quasinormal inGifHpermutes with every Sylow subgroup ofG.A subgroupHofGis said to beS-quasinormally embedded inGif every Sylow subgroup ofHis a Sylow subgroup of someS-quasinormal subgroup ofG.A subgroupHofGis calledc-normal inG(see[21])if there exists a normal subgroupK,such thatG=HKandH∩K≤HG,whereHGis the maximal normal subgroup ofGcontained inH.In[9],Guo et al.gave the concept of ans-embedded subgroup as follows:A subgroupHis said to bes-embedded inGifGhas anS-quasinormal subgroupT,such thatT∩H≤HsGandHT=HsG,whereHsGis the subgroup generated by all those subgroups ofHwhich areS-quasinormal inGandHsGis the intersection of all suchS-quasinormal subgroups ofGwhich containH.By using the above ideas,a series of interesting results was obtained(see,e.g.,[7–11,14–16,21–22]).In this paper,we give some new applications ofS-quasinormal subgroups andS-quasinormally embedded subgroups in the theory of groups.Our main tool here is the following concept.

Definition 1.1Let H be a subgroup of G.We say that H is nearly SS-embedded in G if G has an S-quasinormal subgroup K such that HK is S-quasinormal in G and H∩K≤HseG,where HseGis the subgroup generated by all those subgroups of H which are S-quasinormally embedded in G.

It is easy to see that allS-quasinormal,S-quasinormally embedded,c-normal andsembedded subgroups are all nearlySS-embedded inG.However,the following examples show that the converse is not true.

Example 1.1Suppose thatGis the symmetric groupS4.

(1)LetH=〈(12)〉.It is easy to see thatG=A4HandH∩A4=1.HenceHis nearlySS-embedded inG.However,His clearly notS-quasinormally embedded inG.

(2)LetH=〈(123)〉andK4={(1),(12)(34),(13)(24),(14)(23)}.ThenHK4=A4Gand 1=H∩K4≤HseG.HenceHis a nearlySS-embedded subgroup ofG.But it is easy to check thatHis notc-normal inG.

Example 1.2LetG=S5=A5B,whereB=〈(12)〉,and letK4be the group as in Example 1.1(2).Clearly,K4is a Sylow 2-subgroup ofA5,K4A5=A5GandK4=K4∩A5=(K4)seG.HenceK4is nearlySS-embedded inG,but it is nots-embedded in G.

In this paper,we investigate the inf l uence of the nearlySS-embedded subgroups on the structure of finite groups.Some new results are obtained and some recent results are generalized.

2 Preliminaries

The following known results will be used in this paper.

Lemma 2.1Let G be a group and H≤K≤G.

(1)If H is S-quasinormal in G,then H is S-quasinormal in K(see[13]).

(2)If H is S-quasinormal in G,then H is subnormal in G(see[13]).

(3)If H and L are S-quasinormal in G,then〈H,L〉and H∩L are S-quasinormal in G(see[18]).

(4)If H is S-quasinormal in G and M≤G,then H∩M is S-quasinormal in M(see[3]).

(5)Suppose that H is normal in G.Then K/H is S-quasinormal in G/H if and only if K is S-quasinormal in G(see[13]).

(6)If H is S-quasinormal in G,then H/HGis nilpotent(see[3]).

Lemma 2.2Let A≤K≤G and B≤G.

(1)If A is subnormal in G and B is a minimal normal subgroup of G,then B≤NG(A)(see[4]).

(2)If A is subnormal in G and A is a π-subgroup of G,then A≤Oπ(G).In particular,if A is a subnormal Hall subgroup of G,then A is normal in G(see[23]).

Lemma 2.3(see[10])Suppose that N is a normal subgroup of G and H≤K≤G.Then HseG≤HseKand HseGN/N≤(HN/N)se(G/N).

Lemma 2.4Let H≤K≤G.Then

(1)If H is nearly SS-embedded in G,then H is nearly SS-embedded in K.

(2)Suppose that HG.If K is nearly SS-embedded in G,then K/H is nearly SS-embedded in G/H.

(3)If HG,then for every nearly SS-embedded subgroup E of G with(|H|,|E|)=1,HE/H is nearly SS-embedded in G/H.

Proof(1)Assume that there exists anS-quasinormal subgroupTofG,such thatHTisS-quasinormal inGandH∩T≤HseG.Then by Lemma 2.1(4),T∩KandH(T∩K)=HT∩KareS-quasinormal inK.By Lemma 2.3,H∩(T∩K)≤HseG≤HseK.HenceHis nearlySS-embedded inK.

(2)Assume that there exists anS-quasinormal subgroupTofG,such thatKTisS-quasinormal inGandT∩K≤KseG.SinceHTKisS-quasinormal inGby Lemma 2.1(3),HTK/H=(HT/H)(K/H)isS-quasinormal inG/Hby Lemma 2.1(5).On the other hand,by Lemma 2.3,(HT/H)∩(K/H)=(HT∩K)/H=H(T∩K)/H≤HKseG/H≤(HK/H)se(G/H)=(K/H)se(G/H).HenceK/His nearlySS-embedded inG/H.

(3)Assume thatEis nearlySS-embedded inG.Then there exists anS-quasinormal subgroupTofGsuch thatETisS-quasinormal inGandE∩T≤EseG.By Lemma 2.1(5),(HE/H)(TH/H)=(ET)H/HisS-quasinormal inG/H.Since(|H|,|E|)=1,(|HE∩T:T∩H|,|HE∩T:T∩E|)=1.HenceHE∩T=(T∩H)(T∩E)(see[6,(3.8.1)]).It follows that(HE/H)∩(TH/H)=(HE∩TH)/H=(HE∩T)H/H=(E∩T)H/H≤EseGH/H≤(HE/H)se(G/H)by Lemma 2.3.This shows thatHE/His nearlySS-embedded inG/H.

Lemma 2.5(see[18])Let H be a p-subgroup of G for some prime p.Then H is S-quasinormal in G if and only if Op(G)≤NG(H).

Lemma 2.6(see[1,Lemma 2.4])Let H be a subgroup of G.Then the following two statements are equivalent:

(1)H is an S-quasinormal nilpotent subgroup of G.

(2)The Sylow subgroups of H are S-quasinormal in G.

Lemma 2.7(see[6,(1.8.17)])Let N be a nontrivial solvable normal subgroup of G.If N∩Φ(G)=1,then the Fitting subgroup F(N)of N is the direct product of minimal normal subgroups of G contained in N.

Lemma 2.8Let P be a Sylow p-subgroup of G,where p is the smallest prime dividing|G|.If every maximal subgroup of P is nearly SS-embedded in G,then G is solvable.

ProofSuppose that the assertion is false and letGbe a counterexample of the minimal order.Thenp=2 by Feit-Thompson’s theorem.We now proceed the proof via the following steps.

(1)O2(G)=1.

Assume thatO2(G)/=1.Clearly,P/O2(G)is a Sylow 2-subgroup ofG/O2(G).LetM/O2(G)be a maximal subgroup ofP/O2(G).ThenMis a maximal subgroup ofP.By the hypothesis and Lemma 2.4(2),M/O2(G)is nearlySS-embedded inG/O2(G).The minimal choice ofGimplies thatG/O2(G)is solvable.It follows thatGis solvable.This contradiction shows that(1)holds.

(2)O2′(G)=1.

Suppose thatO2′(G)/=1.ThenPO2′(G)/O2′(G)is a Sylow 2-subgroup ofG/O2′(G).Assume thatM/O2′(G)is a maximal subgroup ofPO2′(G)/O2′(G).Then there exists a maximal subgroupTofP,such thatM=TO2′(G).By the hypothesis and Lemma 2.4(3),M/O2′(G)=TO2′(G)/O2′(G)is nearlySS-embedded inG/O2′(G).The minimal choice ofGimplies thatG/O2′(G)is solvable.By Feit-Thompson’s theorem,we know thatO2′(G)is solvable and so isG,a contradiction.

(3)Pis not cyclic.

IfPis cyclic,thenGis 2-nilpotent by[17,(10.1.9)].This implies thatGis solvable,a contradiction.

(4)If 1/=NG,thenNis not solvable andG=PN.

Suppose thatNis solvable.ThenO2(N)/=1 orO2′(N)/=1.SinceO2(N)charNG,we getO2(N)≤O2(G).Similarly,O2′(N)≤O2′(G).Hence,O2(G)/=1 orO2′(G)/=1,which contradicts(1)or(2).Therefore,Nis not solvable.Assume thatPN＜G.Then by Lemma 2.4(1),every maximal subgroup ofPis nearlySS-embedded inPN.Thus,PNsatisfies the hypothesis.By the choice ofG,we have thatPNis solvable and so isN,a contradiction.Thus(4)holds.

(5)Ghas a unique minimal normal subgroup,and we still denote it byN.

By(4),we see thatG=PNfor every non-identity normal subgroupNofG.It is easy to see thatG/Nis solvable.Since the class of all solvable groups is closed under the subdirect product,Ghas a unique minimal normal subgroup.

(6)Final contradiction.

IfN∩P≤Φ(P),thenNis 2-nilpotent by[12,IV,Theorem 4.7].LetN2′be the normal 2-complement ofN.SinceN2′charNG,we haveN2′ ≤O2′(G).ThusNis a 2-subgroup by(2),soNis solvable.This contradiction shows thatN∩PΦ(P).It follows that there exists a maximal subgroupP1ofP,such thatP=P1(P∩N).By the hypothesis,there exists anS-quasinormal subgroupK,such thatP1KisS-quasinormal inGandP1∩K≤(P1)seG.Suppose that(P1)seG/=1.Let(P1)seG=〈H1,H2,···,Ht〉,whereH1,···,Htare all the nontrivialS-quasinormal embedded subgroups ofGcontained inP1.Then there existS-quasinormal subgroupsK1,K2,···,KtofG,such thatHi∈Syl2(Ki)fori=1,2,···,t.If(Ki)G/=1 for somei∈{1,2,···,t},thenN≤(Ki)G≤Kiby(5).It is easy to see thatHi∩N∈Syl2(N)andHi∩N≤P1∩N≤P∩N∈Syl2(N).HenceHi∩N=P1∩N=P∩N.Consequently,P=(P∩N)P1=(P1∩N)P1=P1,a contradiction.Therefore(Ki)G=1.It follows from Lemma 2.1(6)thatKi=Ki/(Ki)Gis nilpotent andS-quasinormal inG.By Lemma 2.6,HiisS-quasinormal inG.Hence,by Lemma 2.1(3),(P1)seGisS-quasinormal inG,so(P1)seG≤O2(G)=1 by Lemma 2.1(2)and 2.2(2),which implies thatP1∩K=1.IfK=1,thenP1isS-quasinormal inG,soP1is subnormal inGby Lemma 2.1(2).It follows thatP1≤O2(G)=1,soPis cyclic,which contradicts(3).We may,therefore,assume thatK/=1.Clearly,|K|2≤2.ThenKis a 2-nilpotent group by[17,(10.1.9)].LetK2′be a normal Hall 2′-subgroup ofK.SinceKis anS-quasinormal subgroup ofG,Kis subnormal inG,soK2′is subnormal inG.Then by Lemma 2.2(2),K2′ ≤O2′(G)=1.This means thatKis a group of order 2 andP1Kis a Sylow 2-subgroup ofG.SinceP1Kis subnormal inG,P1KGby Lemma 2.2(2),and consequently,N≤P1K=P.This implies thatG=PN=P.The fi nal contradiction completes the proof.

3 Main Results

Theorem 3.1Let P be a Sylow p-subgroup of G,where p is a prime divisor of|G|with(|G|,p?1)=1.If every maximal subgroup of P is nearly SS-embedded in G,then G is pnilpotent.

ProofSuppose that the theorem is false and letGbe a counterexample of the minimal order.Then

(1)Op′(G)=1.

Suppose thatD=Op′(G)/=1.Clearly,PD/Dis a Sylowp-subgroup ofG/Dand every maximal subgroup ofPD/Dmay be written asP1D/D,whereP1is a maximal subgroup ofP.SinceP1is nearlySS-embedded inG,we have thatP1D/Dis also nearlySS-embedded inG/Dby Lemma 2.4(3).ThereforeG/Dsatisfies the condition of the theorem.The minimal choice ofGimplies thatG/Disp-nilpotent and consequentlyGisp-nilpotent,a contradiction.

(2)Gis solvable.

This can be obtained by Lemma 2.8 and the Feit-Thompson theorem.

(3)Ghas a unique minimal normal subgroupN,such thatG=NM,whereMis a nilpotent maximal subgroup ofG,andN=Op(G)=F(G)=CG(N).

LetNbe a minimal normal subgroup ofG.By(1)–(2),Nis an elementary abelianp-group.IfN/=P,then the hypothesis still holds forG/Nby Lemma 2.4(2).The choice ofGimplies thatG/Nisp-nilpotent.IfN=P,thenG/Nis ap′-group and thusG/Nis alsop-nilpotent.Since the class of allp-nilpotent groups is a saturated formation,Nis a unique minimal normal subgroup ofGandNΦ(G).Therefore,there exists a maximal subgroupMofG,such thatNM.It is easy to see thatG=NMandN?Op(G)?F(G)?CG(N).LetC=CG(N).ThenC=C∩NM=N(C∩M).Clearly,C∩MNM=G,which implies thatC∩M=1 and therebyC=N.Hence(3)holds.

(4)The final contradiction.

Obviously,P=P∩NM=N(P∩M),whereP∩M=Mpis a Sylowp-subgroup ofM.LetP1be a maximal subgroup ofPcontainingMp.Clearly,NP1.IfP1=1,then|N|=|P|=p.By(3),G/N～=G/CG(N)is isomorphic with some subgroup of Aut(N),so|G/N|||Aut(N)|.Since|Aut(N)|=p?1 and(|G|,p?1)=1,we have thatG/N=1.Therefore,G=Nis an elementary abelianp-group,a contradiction.

Now suppose thatP1/=1.By the hypothesis,there exists anS-quasinormal subgroupKofG,such thatP1KisS-quasinormal inGandP1∩K≤(P1)seG.Suppose that(P1)seG/=1.LetH1,H2,···,Htbe all the nontrivial subgroups ofP1which areS-quasinormal embedded inG.Then there existS-quasinormal subgroupsK1,K2,···,KtinG,such thatHiis a Sylowpsubgroup ofKifori=1,2,···,t.If(Ki)G/=1 for somei∈{1,2,···,t},thenN≤(Ki)G≤Ki,and thusN≤Hi≤P1,a contradiction.Thus(Ki)G=1 for alli=1,2,···,t.By Lemma 2.1(6),Ki=Ki/(Ki)Gis anS-quasinormal nilpotent subgroup ofG.It follows from Lemma 2.6 thatHiisS-quasinormal inG.Hence(P1)seGisS-quasinormal inGby Lemma 2.1(3).It follows from Lemma 2.2 that(P1)seG≤P1∩Op(G)=P1∩N.Then by Lemma 2.5,This implies thatConsequently,N≤P1,a contradiction.Hence,(P1)seG=P1∩K=1.IfK=1,thenP1isS-quasinormal inG.By Lemma 2.5,N≤(P1)G=(P1)Op(G)P=P1P=P1,a contradiction.Hence,we may assume thatK/=1.Obviously,|Kp|≤p,whereKpis a Sylowp-subgroup ofK.IfKp=1,then,clearly,Kisp-nilpotent.If|Kp|=p,then|Aut(Kp)|=p?1.SinceNK(Kp)/CK(Kp)is isomorphic with some subgroup of Aut(Kp)and(|G|,p?1)=1,we have that|NK(Kp)/CK(Kp)|=1.Hence by Burnside’s Theorem,Kisp-nilpotent.LetKp′be the normalp-complement ofK.ThenKp′is subnormal inG,soKp′≤Op′(G)=1.It follows thatKis a cyclic group of orderpandP1Kis a Sylowp-subgroup ofG.SinceP1KisS-quasinormal inG,P1KGby Lemma 2.2(2).Therefore,P1K=P=Op(G)=Nis an elementary abelianp-group ofG.By Lemma 2.5 and 2.2(1),N≤KG=KOp(G)P=KP=KN=K.It follows thatP1≤KandP1=1.The final contradiction completes the proof.

Corollary 3.1Let p be a prime dividing the order of G with(|G|,p?1)=1and H be a normal subgroup of G,such that G/H is p-nilpotent.If there exists a Sylow p-subgroup P of H,such that every maximal subgroup of P is nearly SS-embedded in G,then G is p-nilpotent.

ProofBy Lemma 2.4(1),every maximal subgroup ofPisSS-embedded inH.By Theorem 3.1,Hisp-nilpotent.Now,letHp′be the normalp-complement ofH.ThenHp′G.Assume thatHp′≠1,and applying Lemma 2.4 again,we see thatG/Hp′satisfies the hypothesis by induction on|G|.HenceG/Hp′isp-nilpotent.It follows thatGisp-nilpotent.We may,therefore,assumeHp′=1.ThenH=Pis ap-group.SinceG/Hisp-nilpotent,we may letK/Hbe the normalp-complement ofG/H.By Schur-Zassenhaus’s theorem,there exists a Hallp′-subgroupKp′ofKsuch thatK=HKp′.Now by using Lemma 2.4(1)and Theorem 3.1,we see thatKisp-nilpotent,soK=H×Kp′.In this case,Kp′is a normalp-complement ofG,and thusGisp-nilpotent.

Corollary 3.2Suppose that every maximal subgroup of any Sylow subgroup of G is nearly SS-embedded in G.Then G is a Sylow tower group of the supersolvable type.

ProofIt follows from Theorem 3.1 and Lemma 2.4.

Theorem 3.2Let p be an odd prime dividing the order of a group G and P be a Sylow psubgroup of G.If NG(P)is p-nilpotent and every maximal subgroup of P is nearly SS-embedded in G,then G is p-nilpotent.

ProofSuppose that the theorem is false and letGbe a counterexample of the minimal order.We proceed via the following steps.

(1)Op′(G)=1.

Suppose thatOp′(G)/=1.Obviously,POp′(G)/Op′(G)is a Sylowp-subgroup ofG/Op′(G).LetT/Op′(G)be a maximal subgroup ofPOp′(G)/Op′(G).ThenT=P1Op′(G)for some maximal subgroupP1ofP.By Lemma 2.4(3)and the hypothesis,P1Op′(G)/Op′(G)is nearlySS-embedded inG/Op′(G).On the other hand,sinceN(G/Op′(G))(POp′(G)/Op′(G))=NG(P)Op′(G)/Op′(G)(see[6,(3.6.10)]),we see thatN(G/Op′(G))(POp′(G)/Op′(G))isp-nilpotent.This shows thatG/Op′(G)satisfies the hypothesis of the theorem.ThusG/Op′(G)isp-nilpotent.It follows thatGisp-nilpotent,a contradiction.

(2)IfMis a proper subgroup ofGcontainingP,thenMisp-nilpotent.

Firstly,clearly,NM(P)isp-nilpotent.By Lemma 2.4(1),we see thatMsatisfies the hypothesis.The minimal choice ofGimplies thatMisp-nilpotent.

(3)G=PQandOp(G)/=1,whereQis a Sylowq-subgroup ofGwithq/=p.

SinceGis notp-nilpotent,by Thompson’s theorem(see[20]),there exists a nonidentity characteristic subgroupHofPsuch thatNG(H)is notp-nilpotent.SinceNG(P)isp-nilpotent,we may choose a characteristic subgroupHofPsuch thatNG(H)is notp-nilpotent,butNG(K)isp-nilpotent for every characteristic subgroupKofPwithH＜K≤P.SinceHcharPNG(P),we haveHNG(P),soNG(P)＜NG(H).Then by(2),we haveG=NG(H).This shows thatH≤Op(G) ≠1 andNG(K)isp-nilpotent for any characteristic subgroupKofPwithOp(G)＜K≤P(if it exists).In this case,using Thompson’s theorem again,we see thatG/Op(G)isp-nilpotent and thenGisp-solvable.Thus for anyq∈π(G)withq≠p,there exists a Sylowq-subgroupQofG,such thatPQis a subgroup ofG(see[5,(6.3.5)]).IfPQ＜G,thenPQisp-nilpotent by(2).It follows from(1)and[17,(9.3.1)]thatQ≤CG(Op(G))≤Op(G),a contradiction.Thus(3)holds.

(4)Ghas a unique minimal normal subgroupN,such thatG=NM,whereMis a maximal subgroup ofG,andN=Op(G)=CG(N).

LetNbe a minimal normal subgroup ofG.Then by(1)and(3),Nis an elementary abelianp-group,andN?Op(G)＜P.It is easy to see thatG/Nsatisfies the hypothesis.HenceG/Nisp-nilpotent by the choice ofG.Since the class of allp-nilpotent groups is a saturated formation,Nis a unique minimal normal subgroup ofGandNΦ(G).Consequently,G=NMfor some maximal subgroupMofG.AsGis solvable by(3),we see thatCG(N)=Op(G)=N.Hence,(4)holds.

(5)Conclusion.

By(4),P=P∩NM=N(P∩M)=NMp,whereMp=P∩Mis a Sylowp-subgroup ofM.IfMp=1,thenP=N,soG=NG(N)=NG(P)isp-nilpotent,a contradiction.Hence,we may assume thatMp/=1.LetP1be a maximal subgroup ofPcontainingMp.ThenP=NP1andP1is nearlySS-embedded inG.Therefore,there exists anS-quasinormal subgroupKofGsuch thatP1KisS-quasinormal inGandP1∩K≤(P1)seG.By using the same argument as in the proof of Theorem 3.1,we get thatP1∩K=1 andK≠1.Hence|Kp|≤p,whereKpis a Sylowp-subgroup ofK.If|Kp|=1,thenKis aq-subgroup.By Lemma 2.1(2)and Lemma 2.2(2),K≤Oq(G),which contradicts(1).Hence|Kp|=p.Suppose thatN∩K=1.SinceGis solvable by(3),the minimal normal subgroupK1ofKis an elementary abelianp-group or aq-group.IfK1is ap-group,thenK1≤Op(G)=Nby Lemma 2.2(2),which contradictsN∩K=1.IfK1is aq-group,thenK1≤Oq(G),which contradicts(1).Hence|N∩K|=p.Suppose thatKG/=1.ThenN≤KG≤Kby(4).Therefore,|N|=|N∩K|=p.Ifq＞p,thenNQisp-nilpotent by[17,(10.1.9)],soQ≤CG(N)=N,a contradiction.Ifq＜p,thenM～=G/N=NG(N)/CG(N)is isomorphic with some subgroup of Aut(N).Since Aut(N)is a cyclic group of orderp?1,we have thatQis cyclic.ThenGisq-nilpotent by[17,(10.1.9)]again and thusPis normal inG.HenceNG(P)=Gisp-nilpotent,a contradiction.We may,therefore,assume thatKG=1.Then by Lemma 2.1(6),Kis nilpotent.Hence by(1),Kis ap-subgroup and|K|=p.This means thatP1Kis a Sylowp-subgroup ofG.SinceP1KisS-quasinormal inG,P=P1KGby Lemma 2.2(2).The final contradiction completes the proof.

Corollary 3.3Let p be a prime dividing the order of G and H be a normal subgroup of G,such that G/H is p-nilpotent.If there exists a Sylow p-subgroup P of H,such that NG(P)is p-nilpotent and every maximal subgroup of P is nearly SS-embedded in G,then G is p-nilpotent.

ProofBy Lemma 2.4(1)and Theorem 3.2,Hisp-nilpotent.LetHp′be a normal Hallp′-subgroup ofH.ThenHp′is normal inG.By using the same argument as that in the proof of Corollary 3.1,we may assumeHp′=1 and thusH=P.In this case,G=NG(P)isp-nilpotent.

Theorem 3.3Let G be a p-solvable group and P be a Sylow p-subgroup of G.If every maximal subgroup of P is nearly SS-embedded in G,then G is p-supersolvable.

ProofAssume that the assertion is false and chooseGto be a counterexample of the minimal order.Then

(1)Op′(G)=1.

See the proof of Theorem 3.1.

(2)Op(G)/=1.

SinceGisp-solvable,(2)holds by(1).

(3)Ghas a unique minimal normal subgroupN,such thatG/Nisp-supersolvable,G=NM,whereMis a maximal subgroup ofG,N=Op(G)=F(G)Φ(G),and|N|＞p.

LetNbe a minimal normal subgroup ofGcontained inOp(G).Then by Lemma 2.4,G/Nsatisfies the condition of the theorem,and the minimal choice ofGimplies thatG/Nisp-supersolvable.If|N|=p,thenGisp-supersolvable,a contradiction.Hence|N|＞p.On the other hand,since the class of allp-supersolvable groups is a saturated formation,Nis the unique minimal normal subgroup ofGcontained inOp(G)andOp(G)=N=F(G)Φ(G)by Lemma 2.7.Thus(3)holds.

(4)Final contradiction.

LetP=NMp,whereMpis a Sylowp-subgroup ofM,andP1is a maximal subgroup ofPcontainingMp.By the hypothesis,there exists anS-quasinormal subgroupKofG,such thatP1KisS-quasinormal inGandP1∩K≤(P1)seG.It is easy to verify that(P1)seG=1 as the proof of Theorem 3.1.Therefore,P1∩K=1 and|Kp|≤p.IfKp=1,thenKis ap′-group andK≤Op′(G)=1 by Lemma 2.2(2).It follows thatP1isS-quasinormal inG.By Lemma 2.5,N≤P1G=P1Op(G)P=P1P=P1,a contradiction.Hence we may assume that|Kp|=p.

Suppose thatN∩K=1.SinceGisp-solvable,the minimal normal subgroupK1ofKis an elementary abelianp-group by(1).Clearly,Kis subnormal inGby Lemma 2.1(2).HenceK1≤Op(G)=Nby Lemma 2.2(2).This contradiction shows that|N∩K|=p.Suppose thatKG≠1.ThenN≤KG≤K,so|N|=|N∩K|=p,which contradicts(3).We may,therefore,assume thatKG=1.ThenK/KG=Kis nilpotent by Lemma 2.1(6).Hence,the Sylow subgroups ofKareS-quasinormal inGby Lemma 2.6.IfKis not ap-group andp/=q∈π(K),thenKq≤Op′(G)by Lemma 2.2(2),which contradicts(1).ThusKq=1 and soKis a group of orderp.SinceP1KisS-quasinormal inG,we have thatP1K=PG,and consequentlyN=P.Hence by Lemma 2.5 and Lemma 2.2(1),N≤KG=KOp(G)P=KOp(G)N=KN=K.It follows that|N|=p.The final contradiction completes the proof.

Corollary 3.4If every maximal subgroup of every Sylow subgroup of G is nearly SS-embedded in G,then G is supersolvable.

ProofIt follows directly from Lemma 2.8 and Theorem 3.3.

4 Some Applications

In the literature,one can find the following special cases of Theorems 3.1–3.3.

Corollary 4.1(see[21])Let G be a finite group.Then G is solvable if every maximal subgroup of G is c-normal in G.

Corollary 4.2(see[11])Let p be the smallest prime dividing the order of G and P be a Sylow p-subgroup of G.If every maximal subgroup of P is c-normal in G,then G is p-nilpotent.

Corollary 4.3(see[22])Let G be a group and p be the prime divisor of|G|with(|G|,p?1)=1.If G has a Sylow p-subgroup P such that every maximal subgroup of P not having a p-nilpotent supplement in G is nearly s-normal in G,then G is p-nilpotent.

Corollary 4.4(see[15])Let P be a Sylow p-subgroup of G,where p is a prime divisor of|G|with(|G|,p?1)=1.If every maximal subgroup of P is c-normal or s-quasinormally embedded in G,then G is p-nilpotent.

Corollary 4.5(see[7])Let p be the smallest prime divisor of|G|and P be a Sylow psubgroup of G.If every maximal subgroup of P is S-embedded in G,then G is p-nilpotent.

Corollary 4.6(see[1])Let G be a group and p be the smallest prime dividing|G|.Then G is p-nilpotent if every maximal subgroup of Sylow p-subgroups of G is S-quasinormally embedded in G.

Corollary 4.7(see[11])Let p be an odd prime dividing the order of a group G and P be a Sylow p-subgroup of G.If NG(P)is p-nilpotent and every maximal subgroup of P is c-normal in G,then G is p-nilpotent.

Corollary 4.8(see[21])If every maximal subgroup of every Sylow subgroup of G is cnormal in G,then G is supersolvable.

Corollary 4.9(see[19])Let G be a finite group with the property that the maximal subgroups of Sylow subgroups are S-quasinormal in G.Then G is supersolvable.

Corollary 4.10(see[2])Let G be a finite group.If each maximal subgroup of Sylow subgroups of G is S-quasinormally embedded in G,then G is supersolvable.

[1]Assad,M.and Heliel,A.A.,OnS-quasinormally embedded subgroups of finite groups,J.Pure Appl.Algebra,165,2001,129–135.

[2]Ballester-Bolinches,A.and Pedraza-Aguilera,M.C.,Suffcient conditions for supersolubility of finite groups,J.Pure Appl.Algebra,127,1998,113–118.

[3]Deskins,W.E.,On quasinormal subgroups of finite groups,Math.Z.,82,1963,125–132.

[4]Doerk,K.and Hawkes,T.,Finite Solvable Groups,Walter de Gruyter,New York,1992.

[5]Gorenstein,D.,Finite Groups,Harper and Row Publishers,New York,Evanston,London,1968.

[6]Guo,W.,The Theory of Class of Groups,Science Press-Kluwer Academic Publishers,Beijing,New York,Dordrecht,Boston,London,2000.

[7]Guo,W.,Lu,Y.and Niu,W.,s-embedded subgroups of finite groups,Algebra and Logic,49(4),2010,293–304.

[8]Guo,W.,Shum,K.P.and Skiba,A.N.,On solubility and supersolubility of some classes of finite groups,Sci.China Ser.A,52,2009,272–286.

[9]Guo,W.and Skiba,A.N.,Finite groups with givens-embedded andn-embedded subgroups,J.Algebra,321,2009,2843–2860.

[10]Guo,W.,Skiba,A.N.and Yang,N.,SE-supplemented subgroups of finite groups,Rend.Sem.Mat.Univ.Padova,129,2013,245-263.

[11]Guo,X.and Shum,K.P.,Onc-normal maximal and minimal subgroups of Sylowp-subgroups of finite groups,Arch.Math.,80,2003,561–569.

[12]Huppert,B.,Endliche Gruppen I,Springer-Verlag,Berlin,Heidelberg,New York,1967.

[13]Kegel,O.,Sylow-Gruppen and subnormalteiler endlicher gruppen,Math.Z.,78,1962,205–221.

[14]Li,J.,Chen,G.and Chen,R.,On weaklys-embedded subgroups of finite groups,Sci.China Math.,54,2011,1899–1908.

[15]Li,S.and Li,Y.,OnS-quasinormal andc-normal subgroups of a finite group,Czechoslovak Math.J.,58(133),2008,1083–1095.

[16]Miao,L.,On weaklys-permutable subgroups of finite groups,Bull.Braz.Math.Soc.New Series,41(2),2010,223–235.

[17]Robinson,D.J.S.,A Course in Theory of Group,Spinger-Verlag,New York,1982.

[18]Schmid,P.,Subgroups permutable with all Sylow subgroups,J.Algebra,207,1998,285–293.

[19]Srinivasan,S.,Two suffcient conditions for supersolvability of finite groups,Israel Journal of Mathematics,35,1980,210–214.

[20]Thompson,J.G.,Normalp-complements for finite groups,J.Algebra,1,1964,43–46.

[21]Wang,Y.,c-normality of groups and its properties,J.Algebra,180,1996,954–965.

[22]Wang,Y.and Guo,W.,Nearlys-normality of groups and its properties,Comm.Algebra,38,2010,3821–3836.

[23]Wielandt,H.,Subnormal Subgroups and Permutation Groups,Lectures Given at the Ohio State University,Columbus,Ohio,1971.