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The Arabian Journal for Science and Engineering, Volume 35, Number 2DNovember 2010, Pages 235–240 ON WEAKLY S-SUPPLEMENTED SUBGROUPS OF FINITE GROUPS A subgroup H of a finite group G is said to be weakly s-supplemented in G if G has a subgroup K such that G ¼ HK and H X K a HsG , where HsG is the subgroup of H generatedby all subgroups of H which are S-permutable in G. We are interested in studying the influence ofthe weakly s-supplemented subgroups of prime power order on the structure of finite groups.
All groups considered in this paper will be finite.
Two subgroups H and K of a group G are said to be permutable if hH; Ki ¼ HK ¼ KH. We say fol- lowing Kegel [13], that a subgroup of a group G is S-permutable (S-quasinormal) in G if it permutes withevery Sylow subgroup of G.
Following Ballester-Bolinches et al. [6], we say that a subgroup H of a group G is c-supplemented in G if there exists a subgroup N of G such that G ¼ HN and H X N a HG, where HG ¼ CoreGðHÞ is thelargest normal subgroup of G contained in H.
In 2007, Skiba [15], introduced the following definition: A subgroup H of a finite group G is said to be weakly s-supplemented in G if G has a subgroup K such that G ¼ HK and H X K a HsG, where HsG isthe subgroup of H generated by all subgroups of H which are S-permutable in G. This concept arisesnaturally as an extension of c-supplementation and S-permutability.
The main goal of this paper is to study the structure of a group under assumption that some subgroups of prime power order are weakly s-supplemented.
In this section, we collect some definitions and results that are needed in the sequel.
For a p-group P, we denote WðPÞ ¼ W1ðPÞ if p > 2 and WðPÞ ¼ hW1ðPÞ; W2ðPÞi if p ¼ 2, where Let G be a group of order pe1 pe2 . . . pen , where p i, 1 a i a n, is a prime and pi > piþ1, for all i, 1 a i a n À 1. Let Pi be a Sylow pi-subgroup of G. We shall say that G possesses an ordered Sylow towerif for every k, P1P2 . . . Pk is a normal subgroup of G: Agrawal [1] defined the generalized center, genzðGÞ, of G to be the subgroup generated by all elements g of G such that hgi is S-permutable in G. The generalized hypercenter, genzyðGÞ, is the largestterm of the series 1 ¼ genz0ðGÞ a genz1ðGÞ ¼ genzðGÞ a genz2ðGÞ a Á Á Á , where genziþ1ðGÞ=genziðGÞ ¼genzðG=genziðGÞÞ, for all i b 0.
Let F be a class of groups. We call F a formation provided that (1) if G A F, then G=N A F, and (2) if G=N1 and G=N2 A F, then G=ðN1 X N2Þ A F for arbitrary normal subgroups N, N1, N2 of G. Each groupG has a smallest normal subgroup N such that G=N A F. This uniquely determined normal subgroup ofG is called the F-residual subgroup of G and will be denoted by G F. A formation F is said to be saturatedif G=FðGÞ A F implies G A F. Throughout this paper, U will denote the class of supersolvable groups.
Clearly, U is a formation. Since a group G is supersolvable if and only if G=FðGÞ is supersolvable [12;VI, p. 713], it follows that U is saturated.
For every prime p, we associate some formation Fð pÞ (Fð pÞ could possibly be empty). We say that F is the local formation, locally defined by fFð pÞg provided G A F if and only if for every prime p dividingjGj and every p-chief factor H=K of G, AutGðH=KÞ A FðpÞ ðAutGðH=KÞ denotes the group of automor- Received May 26, 2008; Accepted November 11, 2009.
2010 Mathematics Subject Classification.
Saturated formation, supersolvable groups, weakly s-supplemented subgroups.
phisms induced by G on H=K and it is isomorphic to G=CGðH=KÞÞ. It is known (see [10; Theorem 4.6,p. 368]) that a formation is saturated if and only if it is local.
Let p be a prime. A group G is said to be strictly p-closed whenever P, a Sylow p-subgroup of G, is normal in G with G=P abelian of exponent dividing p À 1. Every strictly p-closed group is supersolvable(see [7; Theorem 1.9, p. 5]).
We assume throughout that F is a formation, locally defined by the system fFð pÞg of full and inte- grated formation Fð pÞ (that is, SpFðpÞ ¼ FðpÞ J F for all primes p, where Sp is the formation of allfinite p-groups). It is well known (see [10; Theorem 3.7, p. 360]) that for any saturated formation F, thereexists a unique integrated and full system which locally defines F. The formation U of all supersolvablegroups is locally defined by the integrated and full system fUð pÞg, where for every prime p, Uð pÞ is theclass of all strictly p-closed groups (see [7; Theorem 1.9, Theorem 1.4 and Corollary 1.5, pp. 4–5]).
A normal solvable subgroup N of a group G is an F-hypercentral subgroup of G provided N possesses a series of subgroups 1 ¼ N0 p N1 p Á Á Á p Nr ¼ N satisfying (i) every factor Niþ1=Ni is a chief factorof G, and (ii) if Niþ1=Ni has a power order of the prime pi, then G=CGðNiþ1=NiÞ A FðpiÞ (see [11]). Theproduct of all F-hypercentral subgroups of G is again an F-hypercentral subgroup of G, denoted byZFðGÞ and called the F-hypercenter of G.
Lemma 2.1. Let p be the smallest prime divisor of the order of a group G, and let Gp be a Sylow p-subgroup of G. If WðGpÞ a genzyðGÞ, then G is p-nilpotent.
Lemma 2.2. Let G be a group and H a K a G.
(i) If H is weakly s-supplemented in G, then H is weakly s-supplemented in K.
(ii) Suppose that H t G. Then EH=H is weakly s-supplemented in G=H for every weakly s-supplemented in G subgroup E satisfying ðjHj; jEjÞ ¼ 1.
Lemma 2.3. If H is a subgroup of G with jG : Hj ¼ p; where p is the smallest prime divisor of jGj, then His normal in G.
Lemma 2.4. Let F be a saturated formation containing U. Suppose that G is a group with a normal sub-group H such that G=H A F. If every subgroup of H of prime order or order 4 is S-permutable in G, thenG A F.
Lemma 2.5. Let K be a normal subgroup of G such that G=K A F, where F is a saturated formation. IfWðPÞ a ZFðGÞ, where P is a Sylow p-subgroup of K, then G=Op0 ðKÞ A F.
Theorem 3.1. Let p be the smallest prime divisor of the order of a group G. If every subgroup of G of order p or 4 (if p ¼ 2) is weakly s-supplemented in G. Then G is p-nilpotent.
Proof. Assume the result is false and let G be a counterexample of minimal order. Then G is not p-nilpotent and so G contains a minimal non-p-nilpotent subgroup K. By a result of Ito [12; IV, Satz 5.4],K is a minimal non-nilpotent group. Now we apply a result of Schmidt [12; III, Satz 5.2], to conclude thatjKj ¼ pnqm for a prime q 0 p, K has a normal Sylow p-subgroup Kp of exponent p if p > 2 or at most4 if p ¼ 2, and also K has a cyclic non-normal Sylow q-subgroup Kq. Clearly, K ¼ KpKq. We treat thefollowing two cases: ON WEAKLY S-SUPPLEMENTED SUBGROUPS OF FINITE GROUPS If every subgroup of K of order p is S-permutable in K, then W1ðKpÞ ¼ Kp a genzyðGÞ. Hence, K is p- nilpotent by Lemma 2.1, a contradiction. Thus, we may assume that a subgroup H of Kp of order p is notS-permutable in K and so H is not S-permutable in G. By hypothesis, H is weakly s-supplemented in G.
Then H is weakly s-supplemented in K by Lemma 2.2(i). Thus, there exists a subgroup T of K such thatK ¼ HT and H X T a HsK , and since H is not S-permutable in K, we have that H X T ¼ 1. By the min-imality of K, T is p-nilpotent. Hence, T ¼ Tp  Tq, where Tp is a Sylow p-subgroup of T and Tq is aSylow q-subgroup of T . Without loss of generality, we take Tq ¼ Kq, i.e., T ¼ Tp  Kq. Since p is thesmallest prime divisor of jGj and jK : Tj ¼ p, we have T is normal in K by Lemma 2.3, and hence Kqis normal in K, a contradiction.
Hence, p ¼ 2 and K2 of exponent 4. By [12; III, Satz 2.5], K 0 ¼ ZðK and K2=K 0 is a chief factor of K. Then W 2 a ZðK Þ. If every subgroup of K2 of order 4 is S- permutable in K, then W2ðK2Þ ¼ K2 a genzyðKÞ. Hence, K is 2-nilpotent by Lemma 2.1, a contradic-tion. Thus, we may assume that there exists a subgroup L of K2 of order 4 that is not S-permutable inK and so L is not S-permutable in G. By hypothesis, L is weakly s-supplemented in G. Then L is weaklys-supplemented in K by Lemma 2.2(i). Thus, there exists a subgroup M of K such that K ¼ LM andL X M a LsK . Since L is not S-permutable in K and W1ðK2Þ a ZðKÞ, we have that LsK is a normalsubgroup of K of order 2. Put N ¼ LsK M. Clearly, N is a proper subgroup of K. Then K ¼ LN andL X N ¼ LsK ðL X MÞ ¼ LsK . By the minimality of K, N is 2-nilpotent and so N ¼ N2 Â Nq, where N2is a Sylow 2-subgroup of N and Nq is a Sylow q-subgroup of N. Without loss of generality, we takeNq ¼ Kq, i.e., N ¼ N2 Â Kq. Clearly, N2 is a maximal subgroup of K2 and so N2 is a normal subgroupof K. With the same arguments as those used in the proof of Case 1, the group K is 2-nilpotent, acontradiction.
Theorem 3.2. Let F be a saturated formation containing U and let G be a group. Then the following twostatements are equivalent: (i) G A F;(ii) There exists a normal subgroup K of G such that G=K A F and every subgroup of K of prime order or order 4 is weakly s-supplemented in G.
Proof. (i) ) (ii): If G A F, then (ii) is true with K ¼ 1.
(ii) ) (i): By hypothesis and Lemma 2.2(i), every subgroup of K of prime order or order 4 is weakly s-supplemented in K. By repeated applications of Theorem 3.1, the group K has a Sylow tower of super-solvable type. Then Kp is a normal Sylow p-subgroup of K, where Kp is a Sylow p-subgroup of K and p is the largest prime dividing the order of K. Since Kp char K and K is a normal subgroup of G, we have that Kp is a normal subgroup of G and since G=K A F, we have that ðG=KpÞ=ðK=KpÞ G G=K A F. ByLemma 2.1(ii), our hypothesis carries over to G=Kp. Then G=Kp A F by induction on the order of G.
If p ¼ 2, then KpQ is a subgroup of G for every Q A SylðGÞ with ðjQj; 2Þ ¼ 1. By Theorem 3.1,KpQ ¼ Kp  Q and since Kp is a normal subgroup of G, it is easily proved that every subgroup of order2 or order 4 of Kp is S-permutable in G. Then G A F by Lemma 2.4. Thus, we may assume that p > 2.
Clearly, 1 a G F a Kp. We assume that G F 0 1. If every subgroup of order p of G F is S-permutable in G,then G A F by Lemma 2.4, a contradiction as G F 0 1. Now we can assume that there exists a subgroupH of order p of G F not S-permutable in G. By hypothesis, H is weakly s-supplemented in G. Then Ghas a subgroup M such that G ¼ HM and H X M a HsG. Since H is not S-permutable in G, it followsthat H X M ¼ 1. Clearly, G ¼ G FM and jG F : G F X Mj ¼ jG : Mj ¼ jH : H X Mj ¼ jHj ¼ p. Hence,G F X M is a maximal subgroup of G F. Then G F X M is a normal subgroup of G F as G F is a p-group,and since G F X M is a normal subgroup of M, we have that G F X M is a normal subgroup ofG ¼ G FM. Hence, ðG=ðG F X MÞÞ=ðG F=ðG F X MÞÞ G G=G F A F. Clearly, G F=ðG F X MÞ is a cyclicgroup of order p. Then G F=ðG F X MÞ a ZUðG=ðG F X MÞÞ. Since U and F are saturated formationssuch that U J F, we have ZUðG=ðG F X MÞÞ a ZFðG=ðG F X MÞÞ, by [10; Proposition 3.11, p. 362].
Hence, G F=ðG F X MÞ a ZFðG=ðG F X MÞÞ. Applying Lemma 2.5, G=ðG F X MÞ A F, a contradiction asG F is the smallest normal subgroup of G such that G=G F A F. Hence, G F ¼ 1 and so G A F.
4. S o m e a p p l i c a t i o n s o f T h e o r e m 3.2 In the literature, one can find the following special cases of Theorem 3.2.
Corollary 4.1. (Buckley [8]) Let G be a group of odd order. If every subgroup of prime order of G is normalin G, then G is supersolvable.
Corollary 4.2. (Wang [16]) If every subgroup of prime order or order 4 is c-normal in G, then G is super-solvable.
Corollary 4.3. (Shaalan [14]) If every subgroup of prime order or order 4 of G is S-permutable in G, then Gis supersolvable.
Corollary 4.4. (Shaalan [14]) Let K be a normal subgroup of a group G such that G=K is supersolvable, andlet every subgroup of prime order or order 4 of K, S-permutable in G. Then G is supersolvable.
Corollary 4.5. (Ballester-Bolinches and Wang [5]) Let F be a saturated formation containing U. If everysubgroup of prime order or order 4 of G F is c-normal in G, then G A F.
Corollary 4.6. (Ballester-Bolinches and Pedraza [4]) Let F be a saturated formation containing U and let Gbe a group with normal subgroup K such that G=K A F. Assume that a Sylow 2-subgroup of G is abelian. Ifevery subgroup of prime order of K is permutable in G, then G A F.
Corollary 4.7. (Wang et al. [17]) Let F be a saturated formation containing U and let G be a group withnormal subgroup K such that G=K A F. If every subgroup of prime order or order 4 of K is c-supplementedin G, then G A F.
(a) Theorem 3.2 is not true for saturated formations which do not contain U. For example, if = is the saturated formation of all nilpotent groups, then the symmetric group of degree three, S3, is a counter-example.
(b) Theorem 3.2 is not true for non-saturated formation. Let F be the formation composed of all groups G such that G U, the supersolvable residual, is elementary abelian. Clearly, U J F but F is notsaturated. Put G ¼ SLð2; 3Þ and K ¼ ZðGÞ. Then G=K is isomorphic to the alternating group of degreefour, A4, and so G=K A F, but G does not belong to F.
(c) Theorem 3.2 is not true in general if we replace the condition ‘‘prime order or order 4’’ by ‘‘prime order’’, as the following example shows. The class F ¼ <A of groups whose derived subgroup is nilpo-tent is a saturated formation containing U (see [12; VI, 9.1(b)]). Consider the group G ¼ GLð2; 3Þ. Thisgroup has a normal subgroup K isomorphic to the quaternion group of order 8 such that G=K is isomor-phic to symmetric group of order 3. Therefore, G=K G S3 A F. Notice that the unique subgroup of Kwith prime order is ZðKÞ and this is not only a weakly s-supplemented subgroup in G, but even is normalsubgroup in G. But the derived group G 0 ¼ SLð2; 3Þ is not nilpotent and so G B F.
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