Finite p-supersoluble groups with some E-S-supplemented subgroups
CHANGWENLI(*) - XIAOLONGYU(**) - NATANG(z)
ABSTRACT- LetHbe a subgroup of a groupGandHeGdenote the subgroup ofH generated by all those subgroups ofHwhich areS-permutably embedded inG.
His said to beE-S-supplemented inGif there exists a subnormal subgroupTof Gsuch thatGHT andH\THeG. In this paper, we investigate the influ- ence of someE-S-supplemented subgroups on the structure of finite groups.
Some new characterizations ofp-supersoluble groups are obtained.
MATHEMATICSSUBJECTCLASSIFICATION(2010). 20D10, 20D20.
KEYWORDS.S-permutable,E-S-supplemented,p-supersoluble.
1. Introduction
Throughout this paper, all groups are finite. Most of the notation is standard and can be found in [4] and [17].Galways denotes a finite group and pdenotes a prime.Gis said to bep-nilpotent if it has a normalp-complement andGis called ap-supersoluble group if its everyp-chief factor is cyclic. It is
(*) Indirizzo dell'A.: School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, China.
E-mail: [email protected]
Supported by NNSF of China (Grant No. 11401264) and the Priority Academic Program Development of Jiangsu Higher Education Institutions.
(**) Indirizzo dell'A.: School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China.
E-mail: [email protected]
Supported by NNSF of China (Grant No. 11071229) and Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20113402110036).
(z) Indirizzo dell'A.: School of Mathematical Science, Huaiyin Normal Uni- versity, Huaiyin, 223300, China.
E-mail: [email protected]
Supported by Tianyuan foundation (Grant No. 11226285).
easy to see that ifGisp-nilpotent thenGis alsop-supersoluble. In the sequel, Upwill denote the class of allp-supersoluble groups andGUpwill denote the Up-residual ofG, i.e., the intersection of all normal subgroupsNofGwith G=N2 Up.
Recall that a subgroupHofGpermutes with a subgroupKofGifHKis a subgroup ofG;His said to beS-permutable inGifHpermutes with all Sylow subgroups ofG. This property extends normality and permutability and was introduced by Kegel [10] in 1962. It turns out to be useful in es- tablishing results concerning the group structure. On the other hand, the study of the generalizations of S-permutability is also one of the most fruitful research areas. For instance, Ballester-Bolinches and Pedraza- Aguilera [2] said thatHisS-permutably embedded inG, if for each primep dividing the order ofjHj, a Sylowp-subgroup ofHis also a Sylowp-sub- group of some S-permutable subgroup of G. Skiba [20] called that His weaklyS-permutable inGif there is a subnormal subgroupTofGsuch that GHTandH\THsG, whereHsGis the subgroup ofHgenerated by all those subgroups ofHwhich areS-permutable inG. Using these general- ized permutable subgroups, one has given a series of necessary and suffi- cient conditions for nilpotency, supersolubility, existence of Sylow towers and, more generally, for belonging to saturated formations containing some of these classes (see [1, 11, 14, 18, 19]). In order to unify above mentioned subgroups, Li, Zhang and Yi in [12] introduced the following concept:
DEFINITION1.1. LetHbe a subgroup ofGandHeGdenote the subgroup ofHgenerated by all those subgroups of Hwhich areS-permutably em- bedded inG. If there is a subnormal subgroupTofGsuch thatGHTand H\THeG, thenHis calledE-S-supplemented inG.
In [12], the authors improved a nice result of Skiba which gives some conditions under which everyG-chief factor of a normal subgroupEofGis cyclic. In present paper, we continue the work of [12] to study the influence ofE-S-supplemented on the structure of finite groups. More precisely, we study p-supersolubility of finite groups under the assumption that some special subgroups areE-S-supplemented.
2. Preliminaries
LEMMA2.1 ([12, Lemma 2.2]). Suppose that H is E-S-supplemented in G.
(1) If HLG, then H is E-S-supplemented in L.
(2) If N/G and NHG, then H=N is E-S-supplemented in G=N.
(3) If H is a p-subgroup and N is a normalp0-subgroup of G, then HN=N is E-S-supplemented in G=N.
LEMMA2.2 ([14, Lemma 2.3]). Suppose that H is S-permutable in G, and let P be a Sylowp-subgroup of H. If HG1, then P is S-permutable in G.
LEMMA 2.3 ([13, Lemma 2.6]). Assume that L is a nontrivial normal subgroup of G and L\F(G)1. Then the Fitting subgroup F(L)of L lies in Soc(G)and therefore F(L)is the direct product of the minimal normal subgroups of G contained in F(L).
LEMMA2.4 ([3, Theorem 2.1.6]). If G is p-supersoluble and Op0(G)1, then the Sylowp-subgroup of G is normal in G.
3. Main Results
THEOREM3.1. Let p be a fixed prime dividing the order of G and L a p- soluble normal subgroup of G such that G=L is p-supersoluble. If there exists a Sylowp-subgroup P of L such that every maximal subgroup of P is E-S-supplemented in G, then G is p-supersoluble.
PROOF. Suppose that the theorem is false and let G be a counter- example of minimal order. We will derive a contradiction in several steps.
(1)Gisp-soluble.
This follows directly from thep-solubility ofLand thep-supersolubility ofG=L.
(2)Op0(G)1.
If TOp0(G)61, we consider GG=T. Clearly, G=LG=LT (G=L)=(TL=L) is p-supersoluble by the p-supersolubility of G=L, where LLT=T. Let P1P1T=T be a maximal subgroup of PT=T. We may assume thatP1is a maximal subgroup ofP. SinceP1isE-S-supplemented inG, the subgroupP1T=TisE-S-supplemented inG=Tby Lemma 2.1(3).
The minimal choice ofGyields thatGisp-supersoluble, and soGis alsop- supersoluble, a contradiction.
(3) IfN is a minimal normal subgroup ofG, thenN is an elementary abelianp-group.
This follows from Steps (1) and (2).
(4)Ghas a unique minimal normal subgroupNcontained inLsuch that G=N isp-supersoluble.
LetN be a minimal normal subgroup ofGcontained inL. Obviously, NPandP=N is a Sylowp-subgroup ofL=N. LetP1=N be a maximal subgroup ofP=N. ThenP1is a maximal subgroup ofP. By hypothesis,P1is E-S-supplemented in G and so P1=N is E-S-supplemented in G=N by Lemma 2.1(2). Since (G=N)=(L=N)G=Lisp-supersoluble,G=Nsatisfies all the hypotheses of our theorem. It follows thatG=Nisp-supersoluble by the minimality ofG. Noticing that the class of allp-supersoluble groups is a saturated formation, we haveNis the unique minimal normal subgroup of Gcontained inL.
(5)jNj>p.
It is clear by Step (4).
(6) The final contradiction.
IfNis contained in all maximal subgroups ofG, thenNF(G) and so Gisp-supersoluble. This contradiction shows that there exists a maximal subgroupMofGsuch thatGNMandN\M1. LetGpbe a Sylowp- subgroup of G containing P. Then GpN(Gp\M) and Gp\M<Gp. Take a maximal subgroupG1ofGpcontainingGp\Mand setP1G1\P.
Then jP:P1j jP:G1\Pj jPG1:G1j jGp:G1j p and so P1 is a maximal subgroup of P. First, we have N6P1. If not, PP\Gp P\NG1N(G1\P)NP1P1, a contradiction. Secondly, we have N\P161. If not,jN:P1\Nj jNP1:P1j jP:P1j pand soP1\N is a maximal ofN. ThereforejNj p, which contradicts Step (5).
By hypothesis,P1isE-S-supplemented inG. Then there is a subnormal subgroupTofGsuch thatGP1TandP1\T(P1)eG. SincejG:Tjis a power ofpandT/ /G,Op(G)T. We knowG=Op(G) is ap-subgroup, so G=Op(G) is p-supersoluble andG=(N\Op(G))9G=NG=Op(G) is p-su- persoluble. ThenN\Op(G)61. Since N is the minimal subgroup ofG, N\Op(G)N and so NOp(G). It follows that N\P1N\(P1)eG. Obviously, (P1)eG61. LetU1;U2; :::;Usbe all the nontrivial subgroups of P1 which are S-permutably embedded in G. For every i2 f1;2; :::;sg, then there is anS-permutable subgroupKiofGsuch thatUiis a Sylowp- subgroup ofKi. Suppose that for somei2 f1;2; :::;sg, we have (Ki)G61.
Then we can take a minimal normal subgroupDofGsuch thatD(Ki)G. In view of Step (3),Dis ap-group. ThenD(P1)eGP1L. By virtue of Step (4), DN. Consequently, NP1. This contradiction shows
that for all i2 f1;2; :::;sg, we have (Ki)G1. By Lemma 2.2, Ui (i2 f1;2; :::;sg) areS-permutable inG. It follows that (P1)eGisS-permutable in G. Let Gq be any Sylow q-subgroup of G with p6q. Then (P1)eGGqGq(P1)eG. SinceN\P1N\(P1)eGN\(P1)eGGq, we have N\P1 is normalized by Gq. Since PGp\L, we have P/Gp and so P1/Gp. It follows thatN\P1 is normalized byGp. Therefore,N\P1is normal in G. The minimality ofN implies thatN\P11 orNP1, a
contradiction. p
We can chooseLto get some results of special interest. For example, if we chooseLGorLGUp, then we obtain the following criteria forp- supersolubility of groups.
COROLLARY3.2. Let P be a Sylowp-subgroup of a p-soluble G, where p is a fixed prime divisor of jGj. If all maximal subgroups of P are E-S- supplemented in G, then G is p-supersoluble.
COROLLARY 3.3. A p-soluble G is p-supersoluble if and only if all maximal subgroups of any Sylowp-subgroup of GUpare E-S-supplemen- ted in G.
In the following, we shall denote byFp(G) thep-Fitting subgroup ofG.
In fact,Fp(G)Op0p(G).
THEOREM3.4. Let p be a fixed prime dividing the order of G and L a p- soluble normal subgroup of G such that G=L is p-supersoluble. If all maximal subgroups of Fp(L)containing Op0(L)are E-S-supplemented in G, then G is p-supersoluble.
PROOF. Suppose that the theorem is false and let G be a counter- example of minimal order. We will derive a contradiction in several steps.
(1)Op0(L)1.
IfOp0(L)61, we consider the factor groupG=Op0(L). First, (G=Op0(L))=(L=Op0(L))G=L
isp-supersoluble. NowOp0(L=Op0(L))1 andFp(L=Op0(L))Fp(L)=Op0(L).
LetM=Op0(L) be a maximal subgroup ofFp(L=Op0(L)). ThenMis a maximal subgroup ofFp(L) containingOp0(L). SinceMisE-S-supplemented inG, in view of Lemma 2.1(2), M=Op0(L) isE-S-supplemented in G=Op0(L). Thus
G=Op0(L) satisfies the hypotheses of the theorem. The minimality ofGim- plies thatG=Op0(L) isp-supersoluble and soGisp-supersoluble, contrary to the choice ofG.
(2)L\F(G)1.
Write HL\F(G). If H61, we consider the factor group G=H.
By virtue of [9, III, 3.5], we have F(L=H)F(L)=Hand so F(L=H) Op(L)=H. On the other hand, writing K=HOp0(L=H) and letting S be a Hall p0-subgroup of K we have KSH, and by the Frattini argument GKNG(S)HNG(S)NG(S) and S/G. Therefore S1 and Op0(L=H)1. This implies that Fp(L=H)Op(L=H)Op(L)=H Fp(L)=H. If P1=H is a maximal subgroup of Fp(L=H), then P1 is maximal in Fp(L) and, by our hypothesis, it is an E-S-supplemented subgroup of G. Hence P1=H is E-S-supplemented in G=H by Lemma 2.1(2). Now the minimality of G implies that G=H is p-supersoluble and then so is G, which contradicts the choice of G.
(3) The final contradiction.
SinceLisp-soluble andOp0(L)1, we haveCL(Op(L))Op(L) by [5, Theorem 6.3.2]. Now F(L)1 implies that F(L)Op(L) is a nontrivial elementary abelian p-group by [9, III, 4.5]. Thus CL(F(L))F(L). By hypothesis, all maximal subgroups of F(L) are E-S-supplemented in G.
Applying [12, Theorem 1.4], allG-chief factors ofF(L) are cyclic. In view of Lemma 2.3, we have F(L)N1N2 Nr, where Ni is a minimal normal of Gwith orderp. Since for each ithe factor group G=CG(Ni) is isomorphic to some subgroup of Aut(Ni), we have G=CG(Ni) is cyclic.
Consequently,G=CG(Ni) isp-supersoluble. From thep-supersolubility of G=L, it follows thatG=(L\CG(Ni))G=CL(Ni) isp-supersoluble. There- foreG=Tr
i1CL(Ni) isp-supersoluble. In fact, what we really have isG=F(L) isp-supersoluble because
\r
i1
CL(Ni)CL(F(L))F(L):
However allG-chief factors ofF(L) are cyclic of orderpand henceGisp-
supersoluble, a contradiction. p
COROLLARY3.5. Let G be a p-soluble group, where p is a fixed prime divisor of jGj. Then G is p-supersoluble if and only if all maximal subgroups of Fp(GUp)containing Op0(GUp)are E-S-supplemented in G.
THEOREM 3.6. Let p be a prime dividing the order of G and P a Sylowp-subgroup of G. If NG(P) is p-nilpotent and all maximal sub- groups of P are E-S-supplemented in G, then G is p-nilpotent. In par- ticular, G is p-supersoluble.
PROOF. It is easy to see that the theorem holds when p2 by [12, Theorem 5.1], so it suffices to prove the theorem for the case of odd prime.
Suppose that the theorem is false and let G be a counterexample of minimal order.
(1) IfHis a proper subgroup ofGwithPH<G, thenHisp-nilpotent.
It is easy to see thatNH(P)NG(P) and henceNH(P) isp-nilpotent. By Lemma 2.1(1), all maximal subgroups of P are E-S-supplemented in H.
HenceHsatisfies the hypothesis of our theorem. The minimal choice ofG implies thatHisp-nilpotent.
(2)Op0(G)1.
If Op0(G)61, we consider G=Op0(G). Obviously, POp0(G)=Op0(G) is a Sylowp-subgroup ofG=Op0(G). In view of Lemma 2.1(3), it is easy to see that all maximal subgroups of POp0(G)=Op0(G) are E-S-supplemented in G=Op0(G). Since
NG=Op0(G)(POp0(G)=Op0(G))NG(P)Op0(G)=Op0(G)
is p-nilpotent, G=Op0(G) satisfies the hypothesis of our theorem. The minimal choice ofGyields thatG=Op0(G) isp-nilpotent and soGisp-nil- potent, a contradiction.
(3)Op(G)61.
Let J(P) be the Thompson subgroup of P. Then NG(P) NG(Z(J(P)))G. If NG(Z(J(P)))<G, then, in view of Step (1), NG(Z(J(P))) isp-nilpotent and soGisp-nilpotent by [5, Theorem 8.3.1], a contradiction. Hence NG(Z(J(P)))G, which shows that Z(J(P)) is a normal p-subgroup ofGand 1<Op(G)<P.
(4)Gisp-soluble.
It is easy to see that the factor groupG=Op(G) satisfies the hypothesis of our theorem. Now, by the minimality ofG, we see that G=Op(G) isp- nilpotent. Consequently,G=Op(G) isp-soluble and soGisp-soluble.
(5) The final contradiction.
Applying Corollary 3.2,Gisp-supersoluble. In view of Lemma 2.4,Pis normal inG. Therefore,GNG(P) isp-nilpotent, a contradiction. p
COROLLARY3.7. Let p be a prime dividing the order of G and L a normal subgroup of G such that G=L is p-nilpotent. Suppose that there exists a Sylowp-subgroup P of L such that all maximal sub- groups of P are E-S-supplemented in G and NG(P) is p-nilpotent.
Then G is p-nilpotent.
PROOF. It is obvious that NL(P) is p-nilpotent and all maximal subgroups ofPareE-S-supplemented inL. Applying Theorem 3.6,Lis p-nilpotent. Now we have Lp0 is the normal Hall p0-subgroup of L.
Furthermore,Lp0 is a normal subgroup ofG. IfLp061, we consider the factor group G=Lp0. First, (G=Lp0)=(L=Lp0)G=L is p-nilpotent and all maximal subgroups of PLp0=Lp0 are E-S-supplemented in G=Lp0 by Lemma 2.1(3). Secondly,NG=Lp0(PLp0=Lp0)NG(P)Lp0=Lp0 isp-nilpotent.
Hence G=Lp0 satisfies the hypothesis of our corollary. By induction, G=Lp0 isp-nilpotent and soGisp-nilpotent, as desired. Hence we may assumeLp01, i.e.,LP. By hypothesis, NG(P)G isp-nilpotent. p THEOREM3.8. Let L be a normal subgroup of G such that G=L is p- supersoluble, where p is a prime divisor ofjLjwith(p 1;jLj)1. Suppose that for a Sylowp-subgroup P of L, there exists a subgroup D of P such that 1<jDj<jPjand every subgroup H of P with orderjHj jDjand every cyclic subgroup of P with order4 (ifjDj 2and P is a non-abelian2-group) is E-S-supplemented in G. Then G is p-supersoluble.
PROOF. Suppose that this theorem is false and consider a counter- example (G;L) for whichjGkLjis minimal.
(1)Lisp-nilpotent.
By Lemma 2.1, it is easy to see that every subgroupHofPwith order jHj jDjand every cyclic subgroup ofPwith order 4 (ifjDj 2 andPis a non-abelian 2-group) areE-S-supplemented inL. Applying [12, Theorem 1.5], Lisp-nilpotent.
(2)PL.
From Step 1, we know Op0(L) is the normal Hall p0-subgroup of L.
Assume that Op0(L)61. In view of Lemma 2.1, the hypothesis holds for (G=Op0(L);L=Op0(L)). Hence, by the minimal choice of (G;L), the theorem is true for (G=Op0(L);L=Op0(L)) and so G=Op0(L) is p-supersoluble. Con- sequently,Gisp-supersoluble. This contradiction shows that Op0(L)1.
HenceLis a normalp-subgroup ofG.
(3) The final contradiction.
Applying [12, Theorem 1.4], allG-chief factors ofLare cyclic. From the p-supersolubility ofG=L, we haveGisp-supersoluble. p COROLLARY3.9. Let p be a prime divisor ofjGUpjwith(p 1;jGUpj)1.
Suppose that for a Sylowp-subgroup P of GUp, there exists a subgroup D of P such that1<jDj<jPjand every subgroup H of P with orderjHj jDj and every cyclic subgroup of P with order 4 (ifjDj 2 and P is a non- abelian2-group)is E-S-supplemented in G. Then G is p-supersoluble.
COROLLARY3.10. Let p be the smallest prime divisor ofjGj. Suppose that for a Sylowp-subgroup P of GUp, there exists a subgroup D of P such that 1<jDj<jPj and every subgroup H of P with order jHj jDjand every cyclic subgroup of P with order4 (ifjDj 2and P is a non-abelian 2-group)is E-S-supplemented in G. Then G is p-nilpotent.
PROOF. By virtue of Corollary 3.9, G is p-supersoluble. Since p is the smallest prime divisor of jGj, we have G is p-nilpotent by [11,
Lemma 2.8]. p
4. Some Applications
It is clear that all the subgroups, whether they arec-normal subgroups [21], c-normal subgroups [22], S-permutably embedded subgroups or weakly S-permutable subgroups, are all E-S-supplemented subgroups.
Hence the following results are all special cases of our Theorems.
COROLLARY4.1 ([22, Theorem 3.7]). Let p be a prime, G a p-soluble group and H a normal subgroup of G such that G=H is p-supersoluble. If all maximal subgroups of Fp(H)containing Op0(H)are c-normal in G, then G is p-supersoluble.
COROLLARY4.2 ([22, Theorem 3.5]). Let p be a prime, G a p-soluble group and H a normal subgroup of G such that G=H is p-supersoluble. If there exists a Sylowp-subgroup P of H such that every maximal subgroup of P is c-normal in G, then G is p-supersoluble.
COROLLARY4.3 ([16, Theorem 3.1]). Let p be a prime, G a p-soluble group and H a normal subgroup of G such that G=H is p-supersoluble. If there exists a Sylowp-subgroup P of H such that every maximal subgroup of P is c-normal in G, then G is p-supersoluble.
COROLLARY4.4 ([8, Theorem 3.10]). Let p be a prime, G a p-soluble group and H a normal subgroup of G such that G=H is p-supersoluble. If there exists a Sylowp-subgroup P of H such that every maximal subgroup of P is S-permutably embedded in G, then G is p-supersoluble.
COROLLARY4.5 ([15, Theorem 3.3]). Let p be a prime and G a p- soluble group. If there exists a Sylowp-subgroup P of G such that every maximal subgroup of P is weakly S-permutable in G, then G is p- supersoluble.
COROLLARY4.6 ([6, Theorem 3.1]). Let p be an odd prime dividing the order of G and P a Sylowp-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 4.7 ([14, Theorem 3.2]). Let p be a prime dividing the order of G and P a Sylowp-subgroup of G. If NG(P)is p-nilpotent and every maximal subgroup of P is S-permutably embedded in G, then G is p-nilpotent.
5. Some Remarks
REMARK 5.1. The hypothesis that L is p-soluble in Theorems 3.1 and 3.4 cannot be removed. Consider for example the group GA5, the alternating group of degree 5. Clearly 1 is the maximal subgroup of any Sylow 5-subgroup of Gand F5(G)1. However,G is not 5-super- soluble.
REMARK5.2. In proving our Theorem 3.6, the assumption thatNG(P) isp-nilpotent is essential. To illustrate the situation, we may also consider GA5andp5.
REMARK5.3. From [7] or [11], we know if there is a subgroupTofG such thatGHTandH\THeG, thenHis calledSE-supplemented or E-supplemented in G. Obviously, the set of all SE-supplemented sub- groups of a group is wider than the set of all itsE-S-supplemented sub- groups. However, the following example shows that we cannot generalize our Theorems 3.1 and 3.6 using SE-supplemented subgroups: Let AZ3jZ2S3, whereZ3 is a cyclic subgroup of order 3,Z2is a cyclic
subgroup of order 2 and S3 is the symmetric group of degree 3. Let BAoZ3, the regular wreath product of A by Z3. Put GO2(B) hxj (x)3i. ThenG(Z3Z3Z3)jA4andGis soluble, whereA4is the alternating group of degree 4. LetPbe a Sylow 3-subgroup ofG. Then NG(P)Pis 3-nilpotent. It is easy to see that every maximal subgroup of PisSE-supplemented inG. ButGis not 3-supersoluble.
Acknowledgments. The authors would like to thank the referee who read the manuscript carefully and contributed a lot of valuable suggestions and useful comments.
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Manoscritto pervenuto in redazione il 12 luglio 2013.
