Finite Groups with some <span class="mathjax-formula">$s$</span>-Permutably Embedded and Weakly <span class="mathjax-formula">$s$</span>-Permutable Subgroups

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CONFLUENTES MATHEMATICI

Fenfang XIE, Jinjin WANG, Jiayi XIA, and Guo ZHONG

Finite Groups with somes-Permutably Embedded and Weaklys-Permutable Subgroups

Tome 5, n^o1 (2013), p. 93-100.

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5, 1 (2013) 93-100

FINITE GROUPS WITH SOME s-PERMUTABLY EMBEDDED AND WEAKLY s-PERMUTABLE SUBGROUPS

FENFANG XIE, JINJIN WANG, JIAYI XIA, AND GUO ZHONG

Abstract.LetGbe a finite group,p the smallest prime dividing the order ofGandP a Sylow p-subgroup of G with the smallest generator numberd. There is a set M_d(P) = {P₁, P2,· · ·, Pd} of maximal subgroups of P such that Td

i=1Pi = Φ(P). In the present paper, we investigate the structure of a finite group under the assumption that every member ofM_d(P) is eithers-permutably embedded or weaklys-permutable inGto give criteria for a group to bep-supersolvable orp-nilpotent.

1. Introduction

All groups considered in this paper are finite. Terminology and notation em- ployed agree with standard usage, as in Robinson [15].

In this paper, we let M(G) be the set of all maximal subgroups of a group G.

An interesting problem in group theory is to study the influence of the elements ofM(G) on the structure ofG. A classical result in this orientation is attributed to Srinivasan [19]. Srinivasan obtained thatGis supersolvable provided that every member ofM(G) is normal inG. This result has been extensively generalized.

Two subgroupsHandKof a groupGare said to be permutable ifHK =KH. H is said to bes-permutable inGifH permutes with every Sylow subgroup ofG, i.e., HP =P H for any Sylow subgroupP ofG. This concept was introduced by O. H.

Kegel in [9] and has been studied widely by many authors, such as [5, 17]. Recently, Ballester-Bolinches and Pedraza-Aquilera [3] generalizeds-permutable subgroups to s-permutably embedded subgroups. H is said to bes-permutably embedded inG provided every Sylow subgroup of H is a Sylow subgroup of some s-permutable subgroup ofG. On the other hand, Wang [22] introduced the concept ofc-normal subgroups. Applying the c-normality of subgroups, Wang obtained new criteria for supersolvability of groups. More recently, Skiba [19] introduced the concept of weaklys-permutable subgroups. His called a weaklys-permutable subgroup ofGif there exists a subnormal subgroupT ofGsuch thatG=HT andH∩T 6H_sGthe subgroup ofHgenerated by all those subgroups ofH which ares-permutable inG.

Weaklys-permutability covers boths-permutability andc-normality. Skiba applied weaklys-permutability to unify viewpoint for a series of similar problems. LetP be a Sylowp-subgroup ofG.Many authors have studied the influence of the members of Md(P) (see the Definition 2.1) on the structure of G, such as [8, 12, 16, 18].

Now, in this paper we continue these work. Speaking more precisely, the structure of a finite group under some assumptions on thes-permutably embedded or weakly s-permutable subgroups in Md(P), for each prime p,is studied and obtain some sufficient conditions for ap-supersolvable group or ap-nilpotent group.

2. Preliminaries

Definition 2.1([10, Definition 1.1]). — Letdbe the smallest generator number of a p-group P. Let Md(P) = {P1, P2,· · · , Pd} be a subset of M(P) such that Td

i=1Pi= Φ(P), the Frattini subgroup ofP.

Math. classification:20D10, 20D20.

Keywords: weakly s-permutable subgoups; s-permutably embedded subgroups; p-nilpotent groups.

93

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94 Fenfang Xie, Jinjin Wang, Jiayi Xia & Guo Zhong

Since|M(P)|= (p^d−1)/(p−1),|Md(P)|=dand when d→ ∞, ((p^d−1)/(p−1))/d→ ∞.

Hence|M(P)|>>|Md(P)|.

Lemma 2.2 ([19, Lemma 2.10]). — LetU be a weakly s-permutable subgroup ofGandN a normal subgroup ofG. Then:

(1) IfU 6H6G, thenU is weaklys-permutable in H;

(2) IfN 6U, then U/N is weaklys-permutable in G/N;

(3) Letπbe a set of primes,U aπ⁰-subgroup andN aπ-subgroup. ThenU N/N is weaklys-permutable in G/N;

Lemma 2.3 ([3, Lemma 1]). — Suppose that H is an s-permutably embedded subgroup of G, K 6 G and N is a normal subgroup of G. Then we have the following:

(1) IfH 6K, thenH is ans-permutably embedded subgroup ofK.

(2)HN/N is ans-permutably embedded subgroup ofG/N.

Lemma 2.4([4, 9, 17]). — (1) IfH 6K6GandH iss-permutable inG, then H iss-permutable inK.

(2) If both H and K are s-permutable subgroups of G, then both H ∩K and

< H, K >ares-permutable inG.

(3) IfH iss-permutable subgroups of GandNEG, thenHN iss-permutable subgroups ofGandHN/N iss-permutable subgroups ofG/N.

(4) A p-subgroup H ofG is s-permutable inG if and only ifNG(H)>O^p(G) for some primep∈π(G).

(5) IfH iss-permutable inG, thenH is subnormal inG.

Lemma 2.5([24, Lemma 2.8]). — LetGbe a group and letpbe a prime number dividing|G|with(|G|, p−1) = 1.Then

(1) IfN is normal inGof order p, thenN lies inZ(G); (2) IfGhas cyclic Sylowp-subgroups, thenGisp-nilpotent;

(3) IfM is a subgroup ofGwith indexp, thenM is normal inG.

Lemma 2.6 ([7, IV, Satz 4.7]). — If P is a Sylow p-subgroup of GandNEG such thatP∩N 6Φ(P), thenN isp-nilpotent.

Lemma 2.7 ([7, III, Satz 3.3]). — Let G be a group, and let N be a normal subgroup ofGandH6G. IfN 6Φ(H), then N6Φ(G).

Lemma 2.8 ([23, Lemma 2.6]). — Let N be a normal subgroup of a group G(N 6= 1). If N∩Φ(G) = 1, then the Fitting subgroup F(N)of N is the direct product of minimal normal subgroups ofGthat are contained in F(N).

Lemma 2.9 ([14, Lemma 2.1]). — LetGbe a group andH 6G. Then H_sG is the uniquely determined largest s-permutable subgroup ofG contained in H. In particular,N_G(H)6N_G(H_sG).

Lemma 2.10 ([25]). — (1) If A is subnormal in Gand the index |G: A| is a p⁰-number, thenAcontains all Sylowp-subgroups ofG.

(2) IfAis a subnormal Hall subgroup ofG, then Ais normal inG

Lemma 2.11 ([5]). — If H is an s-permutable subgroup of a group G, then H/HG is nilpotent.

Lemma 2.12 ([17]). — For a nilpotent subgroup H of G, the following two statements are equivalent:

(1)H iss-permutable inG.

(2) The Sylow subgroups ofH ares-permutable in G.

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Lemma 2.13 ([2]). — Let P be a Sylow p-subgroup of G, and P₁ a maximal subgroup ofP. Then the following two statements are equivalent:

(1)P₁is normal inG.

(2)P₁iss-permutable in G.

3. Main Results

Theorem 3.1. — LetP be a Sylowp-subgroup of a groupG, wherepis a prime divisor of|G|with(|G|, p−1) = 1. If every member of some fixedM_d(P)is either weaklys-permutable or s-permutably embedded inG, Then Gisp-nilpotent.

Proof. —Assume G is not p-nilpotent and let the theorem is false and G a counter-example of minimal order. We writeMd(P) ={P1,· · · , Pd} . Then, each Pi is either weakly s-permutable or s-permutably embedded in G. Without loss of generality, suppose that 1 6k 6d such that (i) every P_i(1 6i 6k) is weakly s-permutable in G. Then there exists a subnormal subgroup K_i of G such that G = P_iK_i and P_i∩K_i 6 (P_i)_sG. (ii)each P_j(k+ 1 6 j 6 d) is s-permutably embedded in G. Then there exists an s-permutable subgroup M_j 6G such that P_j is a Sylowp-subgroup ofM_j.

Now we prove the theorem by the following several steps.

(1)Op⁰(G) = 1.

Consider the quotient group G/Op⁰(G). Since P Op⁰(G)/Op⁰(G) is a Sylow p- subgroup ofG/Op⁰(G), which is isomorphic toP, soP Op⁰(G)/Op⁰(G) has the same smallest generator numberdasP. Set

Md(P Op⁰(G)/Op⁰(G)) ={P1Op⁰(G)/Op⁰(G),· · · , PdOp⁰(G)/Op⁰(G)}.

Also, eachP_sO_p⁰(G)/O_p⁰(G) fors∈ {1,· · ·, d}is eithers-permutably embedded or weaklys-permutable inG/Op⁰(G) by Lemmas 2.2 and 2.3. Thus,G/Op⁰(G) satisfies the conditions of the theorem. IfOp⁰(G)>1, thenG/Op⁰(G) isp-nilpotent by the choice ofG. It follows that Gitself is p-nilpotent, a contradiction.

(2) (Pi)sGCG and the quotient group G/(Pi)sG is p-nilpotent for every i ∈ {1,2,· · ·, k}.

Since Pi CP, by Lemma 2.9 we have P 6 NG(Pi) 6 NG((Pi)sG), that is, (Pi)sG is normalized by P. Clearly, (Pi)sG is an s-permutable p-group and so O^p(G)6NG((Pi)sG) by Lemma 2.4. Now we can get that (Pi)sGCP O^p(G) =G.

Since G = PiKi and Pi∩Ki 6 (Pi)sG. If Pi ∩Ki < (Pi)sG, let Ti denote the subnormal subgroupK_i(P_i)sG.It follows that

PiTi=PiKi(Pi)sG=Pi(Pi)sGKi=PiKi =G and

Pi∩Ti=Pi∩Ki(Pi)sG= (Pi∩Ki)(Pi)sG= (Pi)sG. Now we can assumeG=PiKi andPi∩Ki= (Pi)sG. Then

G/(Pi)sG=Pi/(Pi)sG·Ki/(Pi)sG. Therefore,

|Ki/(Pi)sG|p=|G:Pi|p=|P :Pi|=p,

i.e., the factor groupKi/(Pi)sG possesses a cyclic Sylow subgroup of order p. By Lemma 2.5, we have thatKi/(Pi)sGisp-nilpotent. SoKi/(Pi)sGhas a Hall normal p⁰-subgroupH/(Pi)sG. Then

H/(Pi)sGCCG/(Pi)sG and H/(Pi)sG∈Hall(G/(Pi)sG).

It follows from Lemma 2.10 thatH/(Pi)sGis a normalp-complement ofG/(Pi)sG. Consequently,G/(Pi)sG isp-nilpotent, as desired.

(3) For everyj∈ {k+ 1, k+ 2,· · · , d}, the factor groupG/(Mj)G isp-nilpotent.

By the definition of an s-permutably embedded subgroup, Pj is a Sylow p- subgroup of the s-permutable subgroup Mj of G. It follows that Mj/(Mj)G is

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s-permutable in G/(M_j)G and M_j/(M_j)G is nilpotent by Lemma 2.11. Hence, we may apply Lemma 2.12 to see that every Sylow subgroup of M_j/(M_j)_G is s- permutable inG/(M_j)_G. Thus,P_j(M_j)_G/(M_j)_G is s-permutable inG/(M_j)_G be- causeP_j(M_j)_G/(M_j)_G is a Sylowp-subgroup ofM_j/(M_j)_G. It follows by Lemma 2.13 thatP_j(M_j)_G/(M_j)_G is normal in G/(M_j)_G. So the core (M_j)_G ofM_j con- tains the Sylowp-subgroupPj of Mj and we have|G/(Mj)G|p =p. We conclude thatG/(Mj)G isp-nilpotent by Lemma 2.5. We have that (3) holds.

(4) LetN = (Tk

i=1(Pi)sG)∩(Td

j=k+1(Mj)G).We haveNEG. Now, we can obtain thatNisp-nilpotent. Consider the subgroupP∩N. Recall thatPj ∈Sylp((Mj)G) andPj is a maximal subgroup ofP. We have

P∩N = (

k

\

i=1

(P_i)_sG)∩(

d

\

j=k+1

((M_j)_G∩P)) =

k

\

i=1

(P_i)_sG∩(

d

\

j=k+1

P_j)6

d

\

s=1

P_s= Φ(P).

ThusP∩N 6Φ(P) andNEP N. It is easy to see thatN isp-nilpotent by Lemma 2.6.

(5)N 6Φ(G).

We know thatN possesses a normal Hall p⁰-subgroup U such that N =NpU, where Np ∈ Sylp(N). Then U is normal inG and U 6Op⁰(G) = 1, so U = 1.

Therefore,N is a normalp-subgroup ofG. Now,N 6P∩N 6Φ(P).We see that N 6Φ(G) by Lemma 2.7.

(6) The final contradiction.

By (2) and (3), G/(P_i)_sG and G/(M_j)_G are p-nilpotent. Hence, G/N is a p- nilpotent. SinceN 6Φ(G), it is easy to see that Gisp-nilpotent, the final contra-

diction. The proof of Theorem 3.1 is now complete.

Corollary 3.2([1, Theorem 3.5]). — Letpbe the smallest prime dividing|G|.

IfP is a Sylowp-subgroup ofGsuch that every member ofM(P)iss-permutable inG, thenGhas a normalp-complement.

Corollary 3.3 ([11, Theorem 3.1]). — Suppose that p ∈ π(G) is such that (|G|, p−1) = 1. LetP be a Sylow p-subgroup of a group G. Assume that every member of M(P) is either c-normal or s-permutably embedded inG. ThenG is p-nilpotent.

Corollary 3.4 ([14, Theorem 3.2]). — LetG be a group and P be a Sylow p-subgroup ofG, wherepis a prime divisor of |G| with(|G|, p−1) = 1. Suppose that every member ofM(P)is weakly s-permutable inG, thenGisp-nilpotent.

Theorem 3.5. — Let G be a group and let P be a Sylowpp-subgroup of G such thatNG(P)isp-nilpotent, wherepis a prime divisor of|G|. If every member in some fixed Md(P)is either weakly s-permutable ors-permutably embedded in G, thenGisp-nilpotent.

Proof. —By Theorem 3.1, it is easy to see that the theorem holds when p= 2.

Assume that the theorem is false and letGbe a counter-example of minimal order.

By the hypotheses of the theorem, denoteMd(P) ={P1, P2,· · ·, Pd}. Then, each Pi is either weakly s-permutable or s-permutably embedded in G. Furthermore, we have

(1)Op⁰(G) = 1.

Consider the quotient group G/O_p0(G). Since P O_p0(G)/O_p0(G) is a Sylow p- subgroup ofG/O_p⁰(G), which is isomorphic toP, soP O_p⁰(G)/O_p⁰(G) has the same smallest generator numberdasP. Set

Md(P Op⁰(G)/Op⁰(G)) ={P1Op⁰(G)/Op⁰(G),· · · , PdOp⁰(G)/Op⁰(G)}.

Also, each PsOp⁰(G)/Op⁰(G) for s ∈ {1,· · · , d} is eithers-permutably embedded or weakly s-permutable in G/Op⁰(G) by Lemmas 2.2 and 2.3. Thus, G/Op⁰(G)

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satisfies the conditions of the theorem. If O_p⁰(G) > 1, then G/O_p⁰(G) is p- nilpotent by the choice ofG. It follows thatGitself isp-nilpotent, a contradiction.

Also, N_G/N(P N/N) = N_G(P)N/N, hence it is p-nilpotent because N_G(P) is p- nilpotent. ThusG/O_p⁰(G) satisfies the hypothesis of our theorem. By the choice ofG, G/O_p⁰(G) isp-nilpotent and it follows thatGisp-nilpotent, a contradiction.

(2) IfP 6H < G, thenH isp-nilpotent.

SinceNH(P)6NG(P),we have thatNH(P) isp-nilpotent. By Lemmas 2.2 and 2.3,H satisfies the hypotheses of the theorem. By the choice ofG,Hisp-nilpotent, as desired.

(3)G=P Q, where Qis a Sylowq-subgroup ofGwithp6=q.

SinceGis notp-nilpotent, by a result of Thompson [21, Corollary], there exists a non-trivial characteristic subgroup T of P such that NG(T) is notp-nilpotent.

Choose T such that the order of T is as large as possible. Since NG(P) is p- nilpotent, we have NG(K) is p-nilpotent for any characteristic subgroup K of P satisfying T < K 6 P. Now, T char P CNG(P), which gives T ENG(P). So NG(P) 6 NG(T). By (2), we have that NG(T) = G and T = Op(G). Now, applying the result of Thompson again, we have that G/O_p(G) isp-nilpotent and thereforeGis p-solvable. Then for anyq∈π(G) withq6=p, there exists a Sylow q-subgroup of Qsuch thatP Q is a subgroup ofG[6,Theorem 6.3.5]. IfP Q < G, thenP Qisp-nilpotent by (2), contrary to the choice ofG. Therefore,P Q=G, as desired.

(4) Every minimal normal subgroup ofGcontained in Op(G) is of orderp.

AsOp⁰(G) = 1, we get that Op(G)>1. Let N be a minimal normal subgroup ofGcontained inOp(G). IfN 6Φ(P),by Lemma 2.7, then N 6Φ(G), andG/N satisfies the hypotheses of the theorem. By the choice ofG,G/N isp-nilpotent. So G/Φ(G) isp-nilpotent, it follows thatGisp-nilpotent, a contradiction. ThusN66 Φ(P). SinceTd

i=1Pi= Φ(P), where Pi∈ Md(P), we can assumeN 66P1without loss of generality. By the conditions of the theorem, P1 is weaklys-permutable in Gors-permutably embedded inG. We claim that|N|=p.

(i) We first consider the case thatP1 is weakly s-permutable inG. Then there exists K1CCG such that G = P1K1 and P1∩K1 6 (P1)sG. Since P1CP, by Lemma 2.9 we have P 6 NG(P1) 6 NG((P1)sG), that is, (P1)sG is normalized byP. Clearly, (P₁)sG is as-permutable p-group and so O^p(G)6N_G((P₁)sG) by Lemma 2.4. Now we can get that (P₁)_sGCP O^p(G) =G. Then (P₁)_sG∩N = 1 or N. If (P₁)_sG∩N =N, thenN 6(P₁)_sG6P₁, a contradiction. So we have that (P₁)_sG∩N = 1, thenP₁∩K₁∩N = 1. From the minimal normality ofN, we know that (K₁)_G∩N = 1 orN. If (K₁)_G∩N = 1, then

N ∼=N(K1)G/(K1)GCG/(K1)G,

where G/(K1)G is a p-group since all Sylow q-subgroups ofGis contained in K1

by Lemma 2.10. Thus we have that |N| = p. If (K1)G ∩N 6= 1, we get that N 6(K1)G6K1. Then

1 =P1∩K1∩N =P1∩N and soN P₁=P. We also get|N|=p.

(ii) Next, we consider that case that P₁ is s-permutably embedded inG. If P₁ is s-permutably embedded in G, then there exists an s-permutable subgroup H such that P₁ ∈ Sylp(H). Hence,HQ is a subgroup of G. Since NCG, we have that N1 =N∩HQCHQ. It follows that N1ChHQ, Ni=G.Moreover, by the minimality normality ofN, we get that N1= 1 and so|N|=p.

Now, we know thatN∩P1= 1. By [7, I, 17.4], there exists a subgroupM ofG such that G=N M andN∩M = 1. Certainly,N 66Φ(G). From Lemma 2.8, we conclude

Op(G) =R1×R2× · · · ×Rt,

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whereR_i(i= 1,· · ·, t) is a normal subgroup of orderp. It follows that P 6

t

\

i=1

CG(Ri) =CG(Op(G)).

Furthermore, according to [15, Theorem 9.31] and (3), we have thatCG(Op(G))6 Op(G) and soP =Op(G). ThusG=NG(P).Now, by the hypotheses thatNG(P) isp-nilpotent, we conclude thatGisp-nilpotent. This is the final contradiction and

the proof is complete.

Corollary 3.6 ([14, Theorem 3.1]). — Letp be an odd prime divisor of |G|

andP be a Sylowp-subgroup ofG. IfNG(P)is p-nilpotent and every member of M(P)is weakly s-permutable inG, thenGisp-nilpotent.

Theorem 3.7. — Let G be a p-solvable group and let P be a Sylow p-sub- group of G, where p is a prime divisor of |G|. If every member in some fixed M_d(P) is either weaklys-permutable or s-permutably embedded inG, then Gis p-supersolvable.

Proof. —Assume that the theorem is false and let G be a counter-example of minimal order. We write Md(P) ={P1,· · · , Pd} . Then, eachPi is either weakly s-permutable ors-permutably embedded inG.

(1)O_p0(G) = 1.

With an argument similar to that above, (1) holds.

(2) Φ(P)_G= 1, in particular, Φ(O_p(G)) = 1.

Otherwise, then let N = Φ(P)_G > 1. We consider the factor group G/N. Obviously,M_d(P/N) ={P₁/N,· · · , P_d/N}. By Lemmas 2.2 and 2.3,P_i/Nis either weakly s-permutable or s-permutably embedded in G/N for any i ∈ {1,· · ·, d}.

Therefore,G/N satisfies the hypotheses of the theorem and consequently,G/N is p-supersolvable by the minimality of G. SinceN 6Φ(P), N 6 Φ(G) by Lemma 2.7, it follows fromG/N being p-supersolvable that Gisp-supersolvable, which is contrary to the choice ofG.

(3) Every minimal normal subgroup ofGcontained in Op(G) is of orderp.

AsOp⁰(G) = 1, we get that Op(G)>1. Let N be a minimal normal subgroup ofGcontained inOp(G). IfN 6Φ(P),by Lemma 2.7, then N 6Φ(G), andG/N satisfies the hypotheses of the theorem. By the choice ofG,G/N isp-supersolvable.

Since the class ofp-supersolvable groups is a saturated formation, we haveGisp- supersolvable, a contradiction. Thus N 66 Φ(P). Since Td

i=1Pi = Φ(P), where Pi∈ Md(P), we can assumeN 66P1 without loss of generality. By the conditions of the theorem, P1 is weaklys-permutable in Gor s-permutably embedded in G.

We claim that|N|=p.

(i) We first consider the case thatP₁ is weakly s-permutable inG. Then there exists K₁CCG such that G = P₁K₁ and P₁∩K₁ 6 (P₁)_sG. Since P₁CP, by Lemma 2.9 we have P 6 N_G(P₁) 6 N_G((P₁)_sG), that is, (P₁)_sG is normalized byP. Clearly, (P₁)_sG is as-permutable p-group and so O^p(G)6N_G((P₁)_sG) by Lemma 2.4. Now we can get that (P₁)_sGCP O^p(G) =G. Then (P₁)_sG∩N = 1 or N. If (P₁)sG∩N =N, thenN 6(P₁)sG6P₁, a contradiction. So we have that (P1)sG∩N = 1, thenP1∩K1∩N = 1. We consider (K1)G∩N. By the minimality ofN, we know that (K1)G∩N = 1 orN. If (K1)G∩N = 1, then

N ∼=N(K₁)_G/(K₁)_GCG/(K₁)_G,

where G/(K₁)G is a p-group since all Sylow q-subgroups ofGis contained in K₁ by Lemma 2.10. Thus we have that |N| = p. If (K₁)_G ∩N 6= 1, we get that N 6(K₁)_G6K₁. Then

1 =P1∩K1∩N =P1∩N and soN P1=P. We also get|N|=p.

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(ii) Next, we consider the case that P₁ is s-permutably embedded in G. If P₁ is s-permutably embedded in G, then there exists an s-permutable subgroup H such that P₁ ∈ Syl_p(H). Hence,HQ is a subgroup of G. Since NCG, we have that N₁ =N∩HQCHQ. It follows that N₁ChHQ, Ni=G.Moreover, by the minimality normality ofN, we get that N₁= 1 and so|N|=p.

Therefore,N∩P₁= 1. By [7, I, 17.4], there exists a subgroupM ofGsuch that G=N M and N∩M = 1. Certainly,N 66Φ(G). Now, we can use Lemma 2.8 to derive thatOp(G) is a direct product of normal subgroups of Gof orderp.

(4) The counter-example does not exist.

SinceG/CG(Ri) is a cyclic group of orderp−1, certainly G/

r

\

i=1

CG(Ri) =G/CG(Op(G))

isp-supersolvable. On the other side, sinceGisp-solvable andOp⁰(G) = 1, by [15, Theorem 9.3.1], C_G(O_p(G))6 O_p(G). Hence, G/O_p(G) is p-supersolvable. Now, claim (3) implies thatGisp-supersolvable. We are done.

Corollary 3.8 ([14, Theorem 3.3]). — LetG be a p-solvable group andP a Sylow p-subgroup ofG. If every member of M(P) is weakly s-permutable in G, thenGisp-supersolvable.

Acknowledgements

The authors are grateful to the referees who provided their valuable suggestions.

This research was supported by NSF of China (10961007, 11161006), NSF of Guangxi (0991101, 0991102), Higher school personnel subsidy scheme of Guangxi (5070).

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Manuscript received October 5, 2012, revised April 10, 2013,

accepted April 16, 2013.

Fenfang XIE, Jinjin WANG, Jiayi XIA & Guo ZHONG

School of Mathematical Sciences, Guangxi Teachers Education University, Nanning, Guangxi, 530023, P. R. China

zhg102003@163.com (Corresponding author)