This article was downloaded by: [University of Illinois Chicago]On: 17 October 2014, At: 04:47Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Communications in AlgebraPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lagb20

On Nearly S-Permutable Subgroups of Finite GroupsKhaled A. Al-Sharo aa Department of Mathematics , Al al-Bayt University , Mafraq , JordanPublished online: 17 Jan 2012.

To cite this article: Khaled A. Al-Sharo (2012) On Nearly S-Permutable Subgroups of Finite Groups, Communications inAlgebra, 40:1, 315-326, DOI: 10.1080/00927872.2010.530329

To link to this article: http://dx.doi.org/10.1080/00927872.2010.530329

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Communications in Algebra®, 40: 315–326, 2012Copyright © Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927872.2010.530329

ON NEARLY S-PERMUTABLE SUBGROUPS OF FINITEGROUPS

Khaled A. Al-SharoDepartment of Mathematics, Al al-Bayt University, Mafraq, Jordan

We say that a subgroup H of a finite group G is nearly S-permutable in G if forevery prime p such that �p� �H�� = 1 and for every subgroup K of G containing H

the normalizer NK�H� contains some Sylow p-subgroup of K. We study the structureof G under the assumption that some subgroups of G are nearly S-permutable in G.

Key Words: Maximal subgroup; Nearly S-permutable subgroup; Saturated formation; Solvablegroup; Supersolvable group; The generalized Fitting subgroup.

2000 Mathematics Subject Classification: 20D10; 20D15.

1. INTRODUCTION

Throughout this article, all groups are finite.An interesting question in finite group theory is to determine the influence of

the embedding properties of members of some families of subgroups on the structureof the group. This article adds some results to this line of research. Let us note inpassing that other interesting generalizations of S-permutability can be found in thearticles [3–6].

Recall that a subgroup H of a group G is said to be S-permutable,S-quasinormal, or ��G�-quasinormal (Kegel [1]) in G if HP = PH for all Sylowsubgroups P of G. S-permutable subgroups possess a series of interesting properties.For instance, if H is an S-permutable subgroup of G, then H is subnormal in G

[1] and H/HG is nilpotent [2]. Moreover, if H ≤ K ≤ G, then H is still S-permutablein K. Therefore, every S-permutable subgroup H of G possesses the followingproperty: if H ≤ K ≤ G and H is S-permutable in G, then NK�H� contains everySylow p-subgroup of K for all primes p such that �p� �H�� = 1. This observation isthe motivation for introducing the following generalization of S-permutability.

Definition 1.1. Let H be a subgroup of a group G. We say that H is nearlyS-permutable in G if for every prime p with �p� �H�� = 1 and for every subgroup K

of G containing H the normalizer NK�H� contains some Sylow p-subgroup of K.

Received July 12, 2010; Revised October 1, 2010. Communicated by A. Turull.Address correspondence to Dr. Khaled A. Al-Sharo, Department of Mathematics, Al al-Bayt

University, Mafraq 25113, Jordan; E-mail: sharo_kh@yahoo.com

315

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

4:47

17

Oct

ober

201

4

316 AL-SHARO

Note that a subgroup of order 6 of the symmetric group S4 on a set with 4letters is nearly S-permutable in S4 and it is not S-permutable in S4. Hence in generalthe set of nearly S-permutable subgroups of a group is wider than the set of all itsS-permutable subgroups.

In this article, we prove the following results.

Theorem A. Let � be a saturated formation containing all supersolvable groups andG a group with normal subgroups X ≤ E such that G/E ∈ � . Suppose that for everySylow subgroup P of X, every maximal subgroup of P or every cyclic subgroup of Pof prime order or order 4 is nearly S-permutable in G. If X is either E or F ∗�E�, thenG ∈ � .

In this theorem F ∗�E� denotes the generalized Fitting subgroup of E, that is,the product of all normal quasinilpotent subgroups of E [7, Chapter X].

Theorem A strengths several results of various authors (see Section 4). Theproof of the theorem consists of a large number of steps. The following theorem isone of the main stages of it.

Theorem B. Let G be a group.

(1) If every minimal subgroup of G is nearly S-permutable in G, then the commutatorsubgroup G′ of G has a normal Sylow 2-subgroup P with nilpotent quotient G′/P.

(2) If for any Sylow subgroup P of G, every maximal subgroup of P or every cyclicsubgroup of P of prime order or order 4 is nearly S-permutable in G, then G issupersolvable.

In [8] Gaschütz described solvable T -groups, that is, groups in which everysubnormal subgroup is normal. Gaschütz results were extended later by Zacher[9] to solvable PT -groups, that is, solvable groups in which every subnormalsubgroup is permutable. As a generalization of these results Agrawal [10] obtaineda description of solvable PST -groups, that is, groups in which every subnormalsubgroup is S-permutable. In this connection it is natural to ask the followingquestion: What can be said about the structure of a solvable group if all of its subnormalsubgroups are nearly S-permutable in the group?

Based on Theorem A, we prove the following theorem.

Theorem C. Let G be a solvable group with every subnormal subgroup of G is nearlyS-permutable in G. Then G = D �M is a supersolvable group, where M is a nilpotentsubgroup, D is a Hall nilpotent subgroup of G of odd order such that G = DNG�H� forevery subgroup H of D. In particular, every maximal subgroup of D is normal in G.

All unexplained notations and terminology are standard. The reader is referredto [11, 12] if necessary.

2. PRELIMINARIES

Recall that a formation � is a homomorph of groups such that each group Ghas the smallest normal subgroup (denoted by G� , and called the � -residual of G)

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

4:47

17

Oct

ober

201

4

ON NEARLY S-PERMUTABLE SUBGROUPS OF FINITE GROUPS 317

whose quotient is still in � . A formation � is said to be saturated if � contains eachgroup G with G/��G� ∈ � .

Lemma 2.1 ([4, Lemma 2.16]). Let � be a saturated formation containing allsupersolvable groups and G a group with a normal subgroup E such that G/E ∈ � . IfE is cyclic, then G ∈ � .

Lemma 2.2. Let H be a nearly S-permutable subgroup of a group G and N a normalsubgroup of G. Then:

(1) HN is nearly S-permutable in G;(2) If H is a group of prime power order, then H ∩ N is nearly S-permutable in G;(3) If H is a group of prime power order, then HN/N is nearly S-permutable in G/N

for any normal subgroup N of G;(4) If �H� = pn for some prime p, then H ≤ Op�G�.

Proof. (1), (2) These assertions follow from NG�H� ≤ NG�HN� and NG�H� ≤NG�H ∩ N�, respectively.

(3) This is evident.

(4) Since H is nearly S-permutable in G and �H� = pn, then �G � NG�H�� =pm for some m. Therefore, G = PNG�H�, where P is a Sylow p-subgroup of Gcontaining H . Hence HG = HPNG�H� = HP ≤ P. Therefore, H ≤ Op�G�.

Lemma 2.3. Let � be a saturated formation containing all nilpotent groups and Ga group with solvable � -residual P = G� . Suppose that every maximal subgroup of Gnot containing P belongs to � . Then P is a p-group for some prime p. In addition, ifevery cyclic subgroup of P of prime order and order 4 (if p = 2 and P is non-abelian)is nearly S-permutable in G, then �P� = p is not the smallest prime divisor of �G�.

Proof. By [13, VI, Theorem 24.2], P = G� is a p-group for some prime p andthe following hold: (1) P/��P� is a � -eccentric chief factor of G (i.e., �P/��P����G/CG�P/��P��� does not belong to � ); (2) P is a group of exponent p or exponent4 (if p = 2 and P is non-abelian); (3) if P is abelian, then ��P� = 1. Let X/��P� bea minimal subgroup of P/��P� such that X/��P� is normal in Gp/��P�, where Gp

is a Sylow p-subgroup of G. Let x ∈ X���P� and L = �x�. Then �L� = p or �L� = 4and so, by hypothesis, L is nearly S-permutable in G. Then X/��P� = L��P�/��P�is nearly S-permutable in G/��P� by Lemma 2.2(3). Hence X/��P� is normal inG/��P� and so X/��P� = P/��P�. Therefore, �P� = p and so �G/CG�P�� dividesp− 1. Finally, note that if p is the smallest prime divisor of �G�, then CG�P� = G andso �P/��P��� �G/CG�P/��P��� P ∈ � . This contradiction completes the proof ofthe lemma.

Lemma 2.4. Let P be a normal p-subgroup of a group G, where p is a prime.Suppose that every cyclic subgroup of P of prime order and order 4 (if P is a non-abelian 2-group) is nearly S-permutable in G. Then every chief factor of G below P iscyclic.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

4:47

17

Oct

ober

201

4

318 AL-SHARO

Proof. First suppose that p > 2. Then P possesses a characteristic subgroup D ofexponent p such that every nontrivial p′-automorphism of P induces a nontrivialautomorphism of D by [16, Chapter 5, Theorem 3.13]. Let H/K be any chief factorof G below D. Let L/K be a minimal subgroup of H/K such that L/K is normal inGp/K, where Gp is a Sylow p-subgroup of G. Let x ∈ L�K. Then �x� = p, so �x� isnearly S-permutable in G. Hence L/K = �x�K/K is nearly S-permutable in G/K byLemma 2.2(3) and so L/K is normal in G/K. Hence H/K = L/K is cyclic. Therefore,every factor of the chief series

1 = D0 < D1 < · · · < Dt−1 = Dt = D (∗)

of G below D is cyclic. Let C0 = C1 ∩ C2 ∩ · · · ∩ Ct, where Ci = CG�Di/Di−1�.Then G/C0 is an abelian group of exponent dividing p− 1 by [15, Chapter 1,Theorem 1.4]. Since C0/CG�D� stabilizes the series �∗�, so C0/CG�D� is a p-groupby [16, Chapter 5, Theorem 3.2]. Since every nontrivial p′-automorphism of P

induces a nontrivial automorphism of D, then CG�D�/CG�P� is a p-group. HenceC0/CG�P� is a p-group, so every chief factor of G below P is cyclic by [15, Chapter 1,Theorem 1.4; Appendixes, Corollary 6.4].

Now suppose that P is a non-abelian 2-group. Let Q be a Sylow q-subgroupof G, where q > 2. Then PQ is nilpotent by Lemma 2.3, so Q ≤ CG�P� and henceO2�G� ≤ CG�P�. But then every chief factor of G below P is cyclic by [15, Chapter 1,Theorem 1.4; Appendixes, Corollary 6.4]. The lemma is proved.

We shall need in our proofs the following known facts about the generalizedFitting subgroup (see Chapter X in [7]).

Lemma 2.5. Let G be a group. Then:

(1) If N is a normal subgroup of G, then F ∗�N� ≤ F ∗�G�;(2) If N is a normal subgroup of G and N ≤ F ∗�G�, then F ∗�G�/N ≤ F ∗�G/N�;(3) F�G� ≤ F ∗�G� = F ∗�F ∗�G��. If F ∗�G� is solvable, then F ∗�G� = F�G�;(4) CG�F

∗�G�� ≤ F�G�.

The following lemma is a direct corollary of [7, X, (13.6)].

Lemma 2.6. Let P be a normal p-subgroup of a group G contained in Z�G�. ThenF ∗�G/P� = F ∗�G�/P.

Lemma 2.7 ([20, Lemma 2.3 (6)]). Let P be a normal p-subgroup of a group G

where p is a prime. Then F ∗�G/��P�� = F ∗�G�/��P�.

Lemma 2.8 ([13, Chapter 2, Lemma 7.9]). Let P be a nilpotent normal subgroup ofa group G. If P ∩��G� = 1, then P is the direct product of some minimal normalsubgroups of G.

Lemma 2.9 ([18, III, Satz 3.5]). Let A�B be normal subgroups of a group G andA ≤ ��G�. Suppose that A ≤ B and B/A is nilpotent. Then B is nilpotent.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

4:47

17

Oct

ober

201

4

ON NEARLY S-PERMUTABLE SUBGROUPS OF FINITE GROUPS 319

3. PROOF OF THEOREM B

Let � be any nonempty set of primes. We use �� to denote the set of allnilpotent �-groups. The symbol � denotes the class of all abelian groups. Recallthat the product �� of the formations � and � is the class �G�G� ∈ ��. It is wellknown that the product of any two formations is also a formation.

Proof of Theorem B. (1) Assume that this assertion is false, and let G be acounterexample with minimal order. Let � be the class of all groups T such thatthe commutator subgroup T ′ of T has a normal Sylow 2-subgroup P with nilpotentquotient T ′/P. Then � = �2�2′�. It is clear that for any set � of primes the set ��

is a formation. Moreover, �� is a saturated formation by [18, Chapter III, Satz 3.5].Hence � is a saturated formation by [29, Corollaries 7.13, 7.19]. Since the hypothesisholds for every subgroup of G, then every maximal subgroup of G belongs to � bythe choice of G.

First we show that G is solvable. Assume that this is false. Then G = G′

and, if F = F�G�, then F = ��G�, G/F is a simple non-abelian group and everyproper normal subgroup of G is contained in F�G�. Let p be the largest primedividing �G�, P = Op�G�, and let H be a subgroup of order p. Then p > 3 by thepaqb-theorem of Burnside and H ≤ P by Lemma 2.2(4). By Lemma 2.4, every chieffactor of G below P is cyclic. Let V be a normal subgroup of G such that V ≤ Pand P/V is a chief factor of G. Then �P/V � = p, so G/CG�P/V� is abelian. HenceCG�P/V� = G since G′ = G. Hence P/V ≤ Z�G/V�. Moreover, P/V ≤ ��G/V� sinceF = ��G�. Therefore, p divides �M�G/F�G���, where M�G/F� is the Schur multiplierof G/F . It is clear that every maximal subgroup of G/F is solvable and hence G/F isisomorphic to L2�q�, L3�3� or to Suzuki group Sz�q� by [18, Chapter II, Remark 7.5].Hence ���M�G/F�G���� ⊆ �2� 3� by [14, Chapter 4]. Therefore, p does not divide�M�G/F�G���. This contradiction shows that G is solvable.

Therefore, G is a solvable minimal non-� -group. Hence by [13, VI, Theorem24.2], G� is a r-group for some prime r. Since � = �2�2′�, r �= 2 and hence �G� � =r by Lemma 2.3. Let M be a maximal subgroup of G such that G = G� M ,and let C = CG�G

� �. Then MG = C ∩M and so G/MG = G�MG/MG�M/MG� issupersolvable, since M/MG G/C is an abelian group. Thus G G/G� ∩MG ∈ � .This contradiction completes the proof of Assertion (1).

(2) Assume that this assertion is false and let G be a counterexample withminimal order. First we prove that G is solvable. Let P be a Sylow p-subgroupof G, where p is the smallest prime dividing �G�. If P is normal in G, then thehypothesis is true for G/P, and so G/P is supersolvable by the choice of G. HenceG is solvable. Therefore, we may assume that P is not normal in G. Suppose thatevery maximal subgroup M of P is nearly S-permutable in G. Then M ≤ Op�G� byLemma 2.2(4). Therefore, since P is not normal in G, P is cyclic. Hence G has anormal Hall p′-subgroup E by [18, IV, Satz 2.8]. The hypothesis holds on E, so Eis supersolvable by the choice of G.

Therefore, G is solvable, so it has a normal subgroup M with prime index �G �M� = r . Let R be a Sylow r-subgroup of G. We shall show that for some prime pdividing �G� a Sylow p-subgroup of G is normal in G. First, assume that every cyclicsubgroup of R with prime order and order 4 is nearly S-permutable in G. Then the

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

4:47

17

Oct

ober

201

4

320 AL-SHARO

hypothesis is true for M , so M is supersolvable by the choice of G. Hence M isp-closed, where p is the largest prime dividing �M�. Let Mp be a Sylow p-subgroup ofM . Then Mp is characteristic in M , and hence, it is normal in G. Therefore, we maysuppose that p = r is the largest prime dividing �G�. It is clear that the hypothesis istrue for G/Mp, so G/Mp is supersolvable by the choice of G. Hence R/Mp is normalG/Mp, which implies the normality of R in G. Now assume that every maximalsubgroup of R is nearly S-permutable in G. Then every maximal subgroup of R isnormal in G. If R is not normal in G, R = �x� is cyclic and every subgroup of �xr�is normal in G. Hence the hypothesis is true for M and for G/�xr�. Hence, as above,we can deduce that some Sylow subgroup P of G is normal in G. It is clear thatthe hypothesis holds for G/P, so G/P is supersolvable by the choice of G. Hencein view of Lemma 2.4 and the choice of G, every maximal subgroup of P is normalin G. Hence the hypothesis is true for G/��P�, so G/��P� is supersolvable by thechoice of G, and hence, G is supersolvable since the class of all supersolvable groupsis a saturated formation. This contradiction completes the proof of Assertion (2).

Corollary 3.1 (Gaschütz [18, IV, Satz 5.7]). If every minimal subgroup of G isnormal in G, then the commutator subgroup G′ of G is 2-closed.

Corollary 3.2 (Buckley [17]). Let G be a group of odd order. If all subgroups of Gof prime order are normal in G, then G is supersolvable.

Corollary 3.3 (Srinivasan [19]). If the maximal subgroups of the Sylow subgroups ofG are normal in G, then G is supersolvable.

4. PROOF OF THEOREM A

Suppose that this theorem is false, and let �G�E� be a counterexample with�G��E� minimal. Let p be the largest prime dividing �X� and P be a Sylow p-subgroupof X. Let C = CG�P�.

(1) X = F ∗�E� �= E.Assume that X = E.

(a) If D is a Hall subgroup of E, the hypothesis is still true for �D�D�. If, inaddition, D is normal in G, then the hypothesis also holds for �G/D�E/D�.

This follows from Lemma 2.2.

(b) If D is a non-identity normal Hall subgroup of E, then D = E.Since D is a characteristic subgroup of E, it is normal in G and by (a) the

hypothesis is still true for �G/D�E/D�. Hence G/D ∈ � , by the choice of G. Thusthe hypothesis is still true for �G�D�, and so D = E, by the choice of �G�E�.

(c) P = E.Indeed, in view of Theorem B, E is supersolvable. Hence P is normal in E, so

P = E by (b).

(d) Every maximal subgroup of P is nearly S-permutable in G and �P� > p.Suppose that this assertion is false. Then every cyclic subgroup of P of prime

order and order 4 (if P is a non-abelian 2-group) is nearly S-permutable in G. Hence

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

4:47

17

Oct

ober

201

4

ON NEARLY S-PERMUTABLE SUBGROUPS OF FINITE GROUPS 321

every chief factor of G below P is cyclic by Lemma 2.4 and so G ∈ � by Lemma 2.1contrary to the choice of G. Hence we have (d).

Let N be a minimal normal subgroup of G contained in P. Then P �= N .Indeed, suppose that P = N . Then some maximal subgroup V of P is normal in aSylow p-subgroup of G, so V is normal in G by (d). Hence V = 1 contrary to (d).Hence P �= N and so the hypothesis is still true for �G/N� P/N�. Therefore, G/N ∈� by the choice of �G�E�. Thus N is the only minimal normal subgroup of Gcontained in P and N � ��G�. Let M be a maximal subgroup of G such that G =NM . Then M ∩ N = 1 and P = P ∩ NM = N�P ∩M�. Since P ≤ F�G� and F�G� ≤CG�N� by [15, Appendixes, Theorem 2.5], it follows that P ∩M is normal in G andso P ∩M = 1. Hence N = P, a contradiction. Therefore, X = F ∗�E� �= E.

(2) F ∗�E/P� �= F ∗/P�Suppose that F ∗�E/P� = F ∗/P� Then the hypothesis holds for �G/P�E/P� and

so G/P ∈ � by the choice of �G�E�. Hence P = E, which contradicts (1).

(3) F ∗ = F and CE�F� = CE�F∗� ≤ F .

The hypothesis is still true for �F ∗� F ∗� and so F ∗ is supersolvable byTheorem B. Hence F ∗ = F by Lemma 2.5(3). Finally, by Lemma 2.5(4), CE�F� =CE�F

∗� ≤ F .

(4) p > 2.Suppose that p = 2. In this case by (3), F ∗ = F is a 2-group. Let Q be

a subgroup of E of prime order q, where q �= 2, and let D = FQ. Then D issupersolvable by (1), and so Q is normal in X. Thus Q ≤ CE�F�. But by (3), CE�F� =CE�F

∗� ≤ F , a contradiction. Hence we have (4).

(5) For some chief factor H/K of G below P, we have �H/K� > p.Suppose that for every chief factor H/K of G below P we have �H/K� = p.

Therefore, every factor of the chief series

1 = P0 < P1 < · · · < Pt−1 = Pt = P (∗)

of G below P is cyclic. Let C0 = C1 ∩ C2 ∩ · · · ∩ Ct, where Ci = CG�Pi/Pi−1�. SinceC0/C stabilizes the series �∗�, C0/C is a p-group by [16, Chapter 5, Theorem 3.2].On the other hand, G/C0 is an abelian group of exponent dividing p− 1, so G/Cis supersolvable by Corollary 1.5 in [15, Chapter 1]. Therefore, G/�C ∩ E� ∈ �since � is a formation containing all supersolvable groups. On the other hand,F ∗�E ∩ C� = F ∗ ∩ C = F ∩ C = V�P ∩ C�, where V is the Hall p′-subgroup of Fby Lemma 2.5(1). Hence the hypothesis holds for �G�E ∩ C�, so E ∩ C = E bythe choice of �G�E�. Hence P ≤ Z�E�, so F ∗�E/P� = F ∗/P by Lemma 2.6, whichcontradicts (2). Therefore, we have (5).

(6) Every maximal subgroup of P is nearly S-permutable G.Suppose that this is false. Then every cyclic subgroup of P of prime order

and order 4 (if P is a non-abelian 2-group) is nearly S-permutable in G. Henceevery chief factor of G below P is cyclic by Lemma 2.4 contrary to (5). Therefore,we have (6).

(7) If L is a minimal normal subgroup of G and L ≤ P, then �L� > p�

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

4:47

17

Oct

ober

201

4

322 AL-SHARO

Assume that �L� = p. Let C0 = CE�L�. Then the hypothesis is true for�G/L�C0/L�. Indeed, clearly, G/C0 = G/�E ∩ CG�L�� ∈ � . Moreover, since L ≤Z�C0� and evidently F = F ∗ ≤ C0 and L ≤ Z�F�, we have F ∗�C0/L� = F ∗/L. On theother hand, if H/L is a maximal subgroup of P/L, then H is maximal in P, so H isnearly S-permutable in G by (6). Therefore, H/L is nearly S-permutable in G/L byLemma 2.2(3). Hence the hypothesis is still true for �G/L�C0/L�. Hence G/L ∈ �and so G ∈ � by Lemma 2.1, a contradiction.

(8) Every proper normal subgroup D of G containing F is supersolvable.By Lemma 2.5(1), F ∗�D� ≤ F ∗ = F ≤ D, and so F ∗�D� = F ∗. Thus the

hypothesis is still true for �D�D�, and so D is supersolvable by the choice of G.

(9) ��G� ∩ P �= 1.Suppose that ��G� ∩ P = 1. Then P is the direct product of some minimal

normal subgroups of G by Lemma 2.8. In view of (6), some maximal subgroup ofP is normal in G, so by [11, Chapter A, 9.13] for some minimal normal subgroup Lof G contained in P, we have �L� = p, which contradicts (7). Thus ��G� ∩ P �= 1.

(10) F ∗�E/L� �= F ∗/L for any nonidentity normal subgroup L of G containedin P.

Suppose that F ∗�E/L� �= F ∗/L� Then the hypothesis holds for �G/L�E/L�, andso G/L ∈ � by the choice of �G�E�. Hence L = P = E, which contradicts (1).

(11) E = G is not solvable.Assume that E is solvable. By (9), ��G� ∩ P �= 1. Let L be a minimal normal

subgroup of G contained in ��G� ∩ P. By Lemma 2.9, F/L = F�E/L�. On the otherhand, F ∗�E/L� = F�E/L�, by Lemma 2.5(3). Hence by (3), F ∗�E/L� = F�E/L� =F ∗/L, which contradicts (10).

(12) G has the only maximal normal subgroup M containing F , M issupersolvable, and G/M is a non-abelian simple group (this directly follows from(8) and (11)).

(13) G/���G� ∩ P) is quasinilpotent.Indeed, in view of (9), ��G� ∩ P �= 1. Hence F/���G� ∩ P� = F ∗/���G� ∩ P� <

F ∗�G/���G� ∩ P�� by (10). But every solvable normal subgroup of F ∗�G/���G� ∩P�� is nilpotent and F/���G� ∩ P� = F�G/���G� ∩ P�� by Lemma 2.9. Hence in viewof (12), F ∗�G/���G� ∩ P�� = G/���G� ∩ P� is quasinilpotent.

(14) G/F is a simple non-abelian group.This follows from (13).

(15) F ∗ = P.Assume that P �= F , and let Q be a Sylow q-subgroup of F , where q �= p. If

G/Q is quasinilpotent, then G G/�P ∩Q� is quasinilpotent, which contradicts (1).Hence F ∗�G/Q� = F/Q, and so the hypothesis hods for �G/Q�G/Q�. Hence G/Q issupersolvable by the choice of G. Hence G is solvable, which contradicts (11).

(16) ��P� = 1.Indeed, F ∗/��P� = F ∗�G/��P�� by Lemma 2.7, so ��P� = 1 by (10).The final contradiction. We show that every chief factor of G below P is

cyclic. First note that in view of (6), some maximal subgroup V of P is normal in

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

4:47

17

Oct

ober

201

4

ON NEARLY S-PERMUTABLE SUBGROUPS OF FINITE GROUPS 323

G. Moreover, in view of (16), every maximal subgroup M of V may be written inthe form M = V ∩D for some maximal subgroup D of P. Hence every maximalsubgroup of V is nearly S-permutable in G by Lemma 2.2(2). Hence by induction,we may suppose that every chief factor of G below V is cyclic contrary to (7). Thetheorem is proved.

We use � to denote the class of all supersolvable groups.In the literature, one can find the following special cases of Theorem A.

Corollary 4.1 (Shaalan [21]). Let G be a group and E a normal subgroup of G withsupersolvable quotient G/E. Suppose that all minimal subgroups of E and all its cyclicsubgroups with order 4 are S-permutable in G. Then G is supersolvable.

Corollary 4.2 (Ballester-Bolinches and Pedraza-Aguilera [22]). Let � be a saturatedformation containing � and G a group with normal subgroup E such that G/E ∈ � .Assume that a Sylow 2-subgroup of G is abelian. If all minimal subgroups of E arequasinormal in G, then G ∈ � .

Corollary 4.3 (Ramadan [23]). Let G be a solvable group. If all maximal subgroupsof the Sylow subgroups of F�G� are normal in G, then G is supersolvable.

Corollary 4.4 (Asaad et al. [24]). Let G be a group and E a solvable normalsubgroup of G with supersolvable quotient G/E. Suppose that all maximal subgroups ofany Sylow subgroup of F�E� are S-permutable in G. Then G is supersolvable.

Corollary 4.5 (Asaad and Csörgo [25]). Let � be a saturated formation containing� and G be a group with a solvable normal subgroup E such that G/E ∈ � . If allminimal subgroups and all cyclic subgroups with order 4 of F�E� are S-permutable inG, then G ∈ � .

Corollary 4.6 (Li and Wang [27]). Let � be a saturated formation containing � andG a group with a normal subgroup E such that G/E ∈ � . If all minimal subgroups andall cyclic subgroups with order 4 of F ∗�E� are S-permutable in G, then G ∈ � .

Corollary 4.7 (Li and Wang [28]). Let � be a saturated formation containing � andG a group with a normal subgroup E such that G/E ∈ � . If all maximal subgroups ofF ∗�E� are S-permutable in G, then G ∈ � .

Corollary 4.8 (Asaad [26]). Let � be a saturated formation containing � and G agroup with a normal subgroup E such that G/E ∈ � . If all maximal subgroups of everySylow subgroup of E are nearly S-permutable in G, then G ∈ � .

5. PROOF OF THEOREM C

The following lemma is well known.

Lemma 5.1. Let G be a group and N � G. Then

�G/N�� = G�N/N�

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

4:47

17

Oct

ober

201

4

324 AL-SHARO

Lemma 5.2. Suppose that G is a metanilpotent group and every subgroup of G� ofprime power order is nearly S-permutable in G. Then G� is a Hall subgroup of G.

Proof. Suppose that this lemma is false, and let G be a counterexample withminimal �G�. Then D = G� �= 1.

Let N be a minimal normal subgroup of G. Then by Lemma 5.1, �G/N�� =G�N/N = DN/N . Let V/N be any subgroup of DN/N of prime power order pa.Then V = N�V ∩D�. If N is a p-group, then V ∩D is a p-group, and so it is nearlyS-permutable in G by hypothesis. Hence V/N is nearly S-permutable in G/N byLemma 2.2. Otherwise, V = NVp, where Vp is a Sylow p-subgroup of V . Hence Vp

is contained in some Sylow p-subgroup of ND, which is a Sylow p-subgroup of D.Hence again V/N is nearly S-permutable in G/N . Therefore, the hypothesis holdsfor G/N , so �G/N�� = DN/N is a Hall subgroup of G/N by the choice of G.

First, assume that G has two minimal normal subgroups H and R such thatH is a p-group and R is a q-group, where p �= q. Then, without loss of generality,we may let H ≤ D. Then DR/R is a Hall subgroup of G/R by the choice of G. LetDp be a Sylow p-subgroup of D. Then RDp/R is a Sylow p-subgroup of DR/R, andso it is a Sylow p-subgroup of G/R. Hence Dp is a Sylow p-subgroup of G. Supposethat Dp �= D, and let Dr be a Sylow r-subgroup of D� where r �= p. Then we see asabove that Dr is a Sylow r-subgroup of G. Thus D is a Hall subgroup of G. Nowwe consider the case when all the minimal normal subgroups of G are p-groups.Then F�G� = Op�G� is a Sylow p-subgroup of G and so D ≤ Op�G�. If H �= D, then,by using the same argument as above, we see that D is a Sylow p-subgroup of G.Hence, we may put H = D. Now we claim that � = ��Op�G�� = 1. In fact, if weassume that � �= 1, then �D/� = �G�/� = �G/��� is a Hall subgroup of G/�.If H ≤ �, then G/� is a nilpotent group. But Op�G� � G, and so � ≤ ��G�. Thisshows that G is nilpotent so that H = G� = 1, a contradiction. Hence H � �, andso H�/� is a nonidentity p-group. Consequently, we have H� = Op�G�. It followsthat H = Op�G�� This contradiction shows that ��Op�G�� = 1. Hence Op�G� is anelementary abelian group. Hence every subgroup of Op�G� is normal in G.

Let Op�G� = �a� × �a2� × · · · × �at�, where �ai� is a minimal normal subgroupof G, �a� = H . Write a1 = aa2 � � � at. Then since �a1� ∩ �a2� � � � �at� = 1, we haveOp�G� = �a1� × �a2� × · · · × �at�. Because G is not nilpotent, Op�G� � Z�G�. Hencethere is an index i such that ai � Z�G�. It is clear that �ai� � G and that�ai� �= H . Since H = D = G� , �ai�H/H ≤ Z�G/H�. Hence from the G-isomorphism�ai�H/H �ai�, we have �ai� ≤ Z�G�, a contradiction. The lemma is proved.

Proof of Theorem C. By Theorem A, G is supersolvable. On the other hand,D = G� is a Hall subgroup of G by Lemma 5.2. Since every chief factor of G iscyclic, it follows that D has odd order. By using the well-known Schur–Zassenhaus’stheorem, we see that D has a complement M in G. Since G/D M is nilpotent, Mis a Hall nilpotent subgroup of G and G = DM .

Finally, we let H be an arbitrary subgroup of D. Let ��H� = �p1� p2� � � � � pt�and Pi be a Sylow pi-subgroup of H and Ni = NG�Pi� for all i = 1� 2� � � � � t. Since Dis nilpotent, H = P1 × P2 × · · · × Pt, so NG�H� ≤ N1 ∩ N2 ∩ · · · ∩ Nt. By hypothesis,�G � Ni� = pi

ai , so �G � NG�H�� = p1b1p2

b2 � � � ptbt . Hence G = DNG�H�. The theorem

is now proved.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

4:47

17

Oct

ober

201

4

ON NEARLY S-PERMUTABLE SUBGROUPS OF FINITE GROUPS 325

REFERENCES

[1] Kegel, O. (1962). Sylow-Gruppen and Subnormalteiler endlicher Gruppen. Math. Z.78:205–221.

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

[3] Ballester-Bolinches, A., Pedraza-Aguilera, M. C. (1998). Sufficient conditions forsupersolvability. J. Pure Appl. Algebra. 127:113–118.

[4] Skiba, A. N. (2007). On weakly s-permutable subgroups of finite groups. J. Algebra315:192–209.

[5] Guo, W., Skiba, A. N. (2009). Finite groups with given s-embedded and n-embeddedsubgroups. J. Algebra 321:2843–2860.

[6] Al-Sharo, K. A., Beidleman, J. C., Heineken, H., Ragland, M. F. (2010). Somecharacterizations of finite groups in which semipermutability is a transitive relation.Forum Math. 22:855–862.

[7] Huppert, B., Blackburn, N. (1982). Finite Groups III. Berlin-Heidelberg-New York:Springer-Verlag.

[8] Gaschütz, W. (1957). Gruppen in denen das Normalteilersein transitiv ist. J. ReineAmgew. Math. 198:87–92.

[9] Zacher, G. (1964). I gruppi risolubili finiti in cue i sottogrupi di compositionecoincidono con i sottogrupi quasi-normali, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis.Mat. Natur. 37(8):150–154.

[10] Agrawal, R. K. (1975). Finite groups whose subnormal subgroups permute with allSylow subgroups. Proc. Amer. Math. Soc. 47(1):77–83.

[11] Doerk, K., Hawkes, T. (1992). Finite Soluable Groups. Berlin–New York: Walter deGruyter.

[12] Ballester-Bolinches, A., Ezquerro, L. M. (2006). Classes of Finite Groups. Dordrecht:Springer.

[13] Shemetkov, L. A. (1978). Formations of Finite Groups. Moscow, Nauka, Main EditorialBoard for Physical and Mathematical Literature.

[14] Gorenstein, D. (1982). Finite Simple Groups. New York–London: Plenum Press.[15] Weinstein, M. (1982). Between Nilpotent and Solvable. Passaic, NJ: Polygonal Publishing

House.[16] Gorenstein, D. (1968). Finite Groups. New York–Evanston–London: Harper & Row

Publishers.[17] Buckley, J. (1970). Finite groups whose minimal subgroups are normal. Math. Z.

15:15–17.[18] Huppert, B. (1967). Endliche Gruppen I. Berlin–Heidelberg–New York: Springer-Verlag.[19] Srinivasan, S. (1980). Two sufficient conditions for supersolvability of finite groups.

Israel J. Math. 35:210–214.[20] Li, D., Guo, X. (1998). The influence of c-normality of subgroups on the structure of

finite groups, II. Comm. Algebra 26:1913–1922.[21] Shaalan, A. (1990). The influence of �-quasinormality of some subgroups on the

structure of a finite group. Acta Math. Hungar. 56:287–293.[22] Ballester-Bolinches, A., Pedraza-Aguilera, M. C. (1996). On minimal subgroups of

finite groups. Acta Math. Hungar. 73:335–342.[23] Ramadan, M. (1992). Influence of normality on maximal subgroups of Sylow

subgroups of a finite group. Acta Math. Hungar. 59:107–110.[24] Asaad, M., Ramadan, M., Shaalan, A. (1991). Influence of �-quasinormality on

maximal subgroups of Sylow subgroups of Fitting subgroups of a finite group. Arch.Math. (Basel) 56:521–527.

[25] Asaad, M., Csörgo, P. (1999). Influence of minimal subgroups on the structure of finitegroup. Arch. Math. (Basel) 72:401–404.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

4:47

17

Oct

ober

201

4

326 AL-SHARO

[26] Asaad, M. (1998). On maximal subgroups of finite group. Comm. Algebra26:3647–3652.

[27] Li, Y., Wang, Y. (2002). The influence of minimal subgroups on the structure of afinite group. Proc. Amer. Math. Soc. 131:337–341.

[28] Li, Y., Wang, Y. (2003). The influence of �-quasinormality of some subgroups of afinite group. Arch. Math. (Basel) 81:245–252.

[29] Shemetkov, L. A., Skiba, A. N. (1978). Formations of Algebraic Systems. Moscow,Nauka, Main Editorial Board for Physical and Mathematical Literature.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

4:47

17

Oct

ober

201

4