Some New Criteria on the Supersolvability of Finite Groups

Int. J. Pure Appl. Sci. Technol., 5(2) (20), pp. 9-30 International Journal of Pure and Applied Sciences and Technology ISSN 2229-607 Available online at www.ijopaasat.in Research Paper Some New Criteria on the Supersolvability of Finite roups Zishan Liu and Shitian Liu 2,,2 School of Science, Sichuan University of Science & Engineering Zigong, 643000, China. Corresponding author, e-mail: (liust@suse.edu.cn) (Received: 27-5-; Accepted: 9-6-) Abstract: In this paper some conditions for supersolvability of finite group are given under the assumption that certain subgroup of is weakly c -normal in a group. Keywords: Weakly c -normality, Supersolvability, s -quasinormally embedded. Introduction The groups considered in this paper are always finite. Ore [6] gave the conception of quasinormality, which is a generalization of normality. A subgroup H is said to be quasinormal in, if for every subgroup K of, then HK = KH. A subgroup H of a group is said to be s -quasinormal (π -quasinormal) in if H permutes with every Sylow subgroup of. This concept was introduced by Kegel in [2], and is extensively studied by Deskins [7]. Ballester-Bolinches and Pedraza-Aguilera [6] introduced the conception of s - quasinormally embedded in, if for each prime divisor p of H, a Sylow p -subgroup of H is also a Sylow p -subgroup of some s -quasinormal subgroup of. Wei and Wang [2] introduced the notion of c -normality, a subgroup H of is

Int. J. Pure Appl. Sci. Technol., 5(2) (20), 9-30 20 said to be c -normal in if there exists a subgroup H K is s -quasinormally embedded in. K such that = HK and In this paper, we give some results by the conception of weakly c -normality. Theorem. Suppose that is a group with a normal subgroup H such that / H is supersolvable. If all maximal subgroups of any Sylow subgroups of H are weakly c -normal in, then is supersolvable. Theorem.2 Suppose that is a group with a normal subgroup H such that / H is supersolvable. If all cyclic subgroups of H with order p or 4 (if p = 2 ) are weakly c -normal in, then is supersolvable. For some notions and notations, please refer to Huppert [9] and Robinson [7]. 2 Some definitions and preliminary results. Definition 2. ([4]) A subgroup H is said to be weakly c -normal in if there exists a subnormal subgroup T of such that = HT and H T H, where H is s -quasinormally embedded subgroup of contained in H. Remark 2. Weakly c-normality and s -quasinormally embedded implies weakly c -normality. Example 2. Every Sylow subgroup of any simple non-abelian group is s - quasinormally embedded but not weakly c -normal. Example 2.2 Let S4 =, the symmetric group of degree 4. Let a = ( 34), then = a A 4. Denote a by P 0. If P be a Sylow 2-subgroup of A 4, P is weakly c -normal but not s -quasinormally embedded in. If P is a Sylow 2-subgroup of some s -quasinormal subgroup K of, then PQ is a subgroup of, where Q be a Sylow 3-subgroup of. Since P0 <, P0 PQ = QP0 P = P = QP, where P = P0 P

Int. J. Pure Appl. Sci. Technol., 5(2) (20), 9-30 2 is a Sylow 2-subgroup. Namely, Q is s-quasinormal in. By [2, Hilfssatz 7], Q is normal in and so is 2-nilpotent, a contradiction. Lemma 2. ([6, Lemma ]) Suppose that A is s -quasinormally embedded in a group, and that H and K <. ()If U H, then U is s -quasinormally embedded in H. (2)If UK is s -quasinormally embedded in, then UK / K is s -quasiormally embedded in / K. (3)If K H and H / K is s -quasinormally embedded in / K, then H is s -quasinormally embedded in. Lemma 2.2 ([4, Lemma 2.2]) Let be a group. Then the following statements hold. ()Let H is weakly c -normal in and H M K. Then H is weakly c -normal in M. (2)Let N < and N H. Then H is weakly c -normal in if and only if H / N is weakly c -normal in / N. (3)Let π be a set of primes. H is a π -subgroup of and N a normal π -subgroup of, if H is weakly c -normal in, then HN / N is weakly c -normal in / N. (4)Let L and H Φ ( L).If H is weakly c -normal in, then H is s -quasinormally embedded in. (5)Let H is c -normal in. Then H is weakly c -normal in. Lemma 2.3 Let M be a maximal subgroup of and P a normal Sylow p -subgroup of such that subgroup of. = PM, where p is a prime. Then P M is a normal Lemma 2.4 ([2, Lemma 2.5]) Let be a group, K an s -quasinormal subgroup of, P a Sylow p -subgroup of K, where p is a prime divisor of. If either p ( ) P O or K =, then P is s -quasinormal in. Lemma 2.5 ([5, Lemma 2.2] Let be a group and P a s-quasinormal p -subgroup

Int. J. Pure Appl. Sci. Technol., 5(2) (20), 9-30 22 of, where p is a prime. Then O p ( ) N ( P). Lemma 2.6 ([2, Lemma 2.8]) Let be a group and p a prime dividing with (, p ) =. ()If N is normal in of order p, then N is in Z ( ). (2)If has cyclic Sylow p -subgroups, then is p -nilpotent. (3)If M and : M = p, then M <. Lemma 2.7 ([8, Satz ]) Suppose that is a group which is not supersolvable but whose proper subgroups are all supersolvable. Then () has a normal Sylow p -subgroup P for some prime p. (2) P / Φ ( P) is a minimal normal subgroup of / ( P) (3)If p 2, then exp( P) = p. (4)If P is non-abelian and p = 2, then exp( P ) = 4. (5)If P is abelian, then exp( P) = p. Lemma 2.8 Let be a group and p a prime number. Φ. ()If P is a minimal normal p-subgroup of, and x P is weakly c -normal in, then P = x. (2)Let P be a normal p-subgroup of and x be an element of P Φ ( P). If P / Φ ( P) is a minimal normal subgroup of / ( P) Φ and x is weakly c -normal in, then P = x. Proof. () Since x is weakly c -normal in, there exists a subnormal subgroup K such that and = x K and x K is s -quasinormally embedded in = PK. Let P = P K. Since P is a minimal normal subgroup of, then P is either trivial or P. If P = P = x P K = x. Otherwise P = P P =, then ( )

Int. J. Pure Appl. Sci. Technol., 5(2) (20), 9-30 23 and hence P K, x = x P = x k x. So x is a normal subgroup of and hence P = x. (2) Since x is weakly c -normal in, then there exists a subnormal subgroup K such that = x K and x K x, where x is an s -quasinormally embedded subgroup of. Let P = P K, Hence P < and PΦ ( P) Φ ( P) is / normal in / Φ ( P). Since P / Φ ( P) is a minimal normal subgroup of / ( P) Φ, Φ ( ) Φ ( ) is either trivial or P / Φ ( P). If the former, then P ( P) P P P / and P = P = x ( P K ) = x Φ ( P) = x. Otherwise P Φ, = P and P K, = PK = K. Hence x = x P = x K x, So x is a normal subgroup of and hence P = x. The Lemma is proved. Lemma 2.9 Let H be a subgroup of. Then H is weakly c -normal in if and only if there exists a subgroup K such that = HK and H K = H. Proof. It is clear. By Definition 2., there exists a subnormal subgroup L of such that = HL and H L H. If H L < H, note that K = LH,then HK = HLH = HL = and H K = H LH = H H L < H. hence ( ) The proof of Main results and their applications The proof of Theorem. Proof. Suppose that the theorem is false, we chose a minimal order group as a counterexample. We will prove the theorem by the following steps: Step. Every proper subgroup of containing H is supersolvable and is solvable. Let N is a proper subgroup of containing H. Then N / H is supersolvable since / H is supersolvable. By hypothesis, all maximal subgroups of any Sylow subgroups of H are weakly normal in, then all maximal subgroups of any Sylow

Int. J. Pure Appl. Sci. Technol., 5(2) (20), 9-30 24 subgroups of H are weakly c - normal in M by Lemma 2.2(). So N, H satisfies the hypotheses of the theorem, the minimal choice of implies that N is supersolvable. Since every maximal subgroup of is supersolvable, then by [8, Hilfssatz C] is solvable. The following. Let L be a minimal normal subgroup of contained in H. Then, by Step and [, Lemma 8. 6, p02] L is an elementary abellian p-group for some prime divisor p of. Step 2. / L is supersolvable and L is unique. / L / H / L / H is supersolvable, and, by hypotheses, all maximal Since ( ) ( ) subgroups of any Sylow subgroups of H are weakly c -normal in, then all maximal subgroups of any Sylow subgroups of H / L are weakly c -normal in / L by Lemma 2.2(2). Then / L, H / L satisfies the hypotheses of the theorem. So the minimal choice of implies that =L is supersolvable. Since the class of supersolvable groups is a saturated formation by [0, Satz 0], then L is a unique minimal normal subgroup of. Step 3. Φ ( ) =. If Φ ( ), then there exists a Sylow p -subgroup P of Φ ( ) divisor of Φ ( ), P char ( ) supersolvable, then / Φ ( ) Φ ( ) =. Step 4. H If H =., where p is a prime Φ and so P is normal in. By Step 2, / P is is supersolvable and so is, a contradiction. Thus <, then H is supersolvable by Step. Let P2 be a Sylow p -subgroup of H, where p is the largest prime divisor of H. Then by [9, VI-9.], P2 a maximal subgroup of P 2. Then, by hypothesis, P is weakly < H. Let P be c -normal in and so there exists a subnormal subgroup T of such that = PT and P T is s -quasinormally embedded in. Let S be a Sylow p -subgroup of P T. Then S is also a Sylow p -subgroup of some s -quasinormal subgroup K of. Let Q be a Sylow

Int. J. Pure Appl. Sci. Technol., 5(2) (20), 9-30 25 q -subgroup of, where q p is a prime divisor of. Then Q is also a Sylow q -subgroup of T. If S =, then by [3, Lemma 2.(3)], then S is π -quasinormal in. and so SQ = QS T. QS is supersolvable by Step and S < QS. Thus we have for every q p, the Sylow q -subgroups commutes with S, then S is s -quasinormal in. Then by Lemma 2.5, we have O p ( ) N ( P) and so S is normal in. If S, then S < S, then S is π -quasinormal. And so QS = SQ T. Obviously, T <, then by Step, QS is supersoluble, and S p Lemma 2.5, we have O ( ) N ( S ) and so S is normal in. < QS. Then by Thus S = L = S is a minimal normal subgroup of by Step 2. By Step 3, F ( H ) is the direct product of minimal normal subgrroup of contained in H by [5, Lemma 2.6], then F ( H ) S L C ( L) = = = since L is the minimally unique normal subgroup of and is solvable. By Lemma 2.7, exp( P) = p or 4 (if p = 2 ), then L is a cyclic subgroup of order p or 4. So is supersolvable by [9, 2.6]. So we assume thatqs H T < H. If H > T, then = PT = PH = H is supersolvable, and so. By Step, T is a supersolvable group. Let R be a Hall p -subgroup of T which is also a Hall p -subgroup of. Then T : R S = p or S =. In the two cases, is supersolvable, a contradiction. Step 5. The final conclusion. = p, so R< T. So we have By Step 4, we have F ( H ) = F ( ). By Step and [5, Lemma 2.6], ( ) F H is the direct products of minimal normal subgroups of. Then F ( H ) F ( H ) L C ( L) = = = as L is abelian and L is the unique minimal normal H

Int. J. Pure Appl. Sci. Technol., 5(2) (20), 9-30 26 subgroup of. Then we have ( ) F H = x, x2, L, xn, where xi char L and xi L \{ }, and so F ( H ) = x = L by Lemma 2.8. By Step 2, / L is supersolvable i and by Step and Lemma 2.7, exp( L) = p or 4 (when p = 2 ), xi is a cyclic subgroup of order p or 4 (when p = 2 ), and so by [9, 2.6], is supersolvable. The final contradiction. This completes the proof. Remark 3. The condition of the theorem / H is supersoluble can't be replaced by / H is soluble. Let = C2 A4, where C2 is a cyclic group of order 2 and A4 is the alternating group of degree 4. Obviously / C2 A4 is soluble. Since every maximal subgroup of A 4 is weakly but is not supersoluble. c -normal in the hypotheses of the theorem is satisfied, Corollary 3. ([2, Theorem 3.])Let L be a complete set of Sylow subgroups of a group. If the maximal subgroups of p L, then is supersoluble. p are L -permuatble subgroups of, for all Corollary 3.2 ([3, Theorem 4. ]) If / H is supersoluble and all maximal subgroups of any Sylow subgroup of H are π -quasinormal in, then is supersoluble. Corollary 3.3 ([2, Theorem 4.]) Suppose that is a group with a normal subgroup H such that / H is supersolvable. If all maximal subgroup of any Sylow subgroup of H is c -normal in, then is supersolvable. The proof of Theorem.2 Proof. Suppose that the theorem is false, we chose a minimal order group as a counterexample. We will prove the theorem by the following steps: Step. Every proper subgroup of containing H is supersolvable and is solvable. Let M be a proper subgroup of containing H. M / H is supersolvable Since / H is supersolvable. By hypothesis, all cyclic subgroups of H of order p or 4 (if p = 2 ) are weakly c -normal in, then by Lemma 2.2(), all cyclic subgroups of H of order p or 4 (if p = 2 ) are weakly hypotheses of the theorem. c -normal in M. Thus (M, H ) satisfies the

Int. J. Pure Appl. Sci. Technol., 5(2) (20), 9-30 27 The minimal choice of implies that M is supersolvable. Then is not supersolvable but all proper subgroups are supersolvable, so by Lemma 2.7, we have has a unique normal Sylow p -subgroup P of for some prime divisor p of. And also is solvable by [8, Hilfssatz C ]. The following. Let L be a minimal normal subgroup of contained in H. Then, by Step and [, Lemma 8.6, p02] L is an elementary abellian p -group for some prime divisor p of. Step 2. / L is supersolvable and L Φ(). By hypothesis, all cyclic subgroups of H of order p or 4 (if p = 2 ) are weakly c -normal in, then by Lemma 2.2(2), all cyclic subgroups of H / L of order p or 4 (if p = 2 ) are weakly c -normal in / / L / H / L / H is L. And since ( ) ( ) supersolvable, then / L, H / L satisfies the hypotheses of the theorem. Thus the minimality of implies that / L is supersolvable. Since the class of supersolvable groups is a saturated formation by [0, Satz 0], then L is unique. If L Φ(), then / Φ ( ) is supersolvable. Thus is supersolvable, a contradiction. Step 3. H = and so F ( H ) = F ( ). If H <, then H is supersolvable by step. Let Q be a Sylow q-subgroup of H, where q is the largest prime divisor of H. Then by [9, VI-9.], Q char H and H <, so Q<. If q p, then / / P / Q is supersolvable, and so is supersolvable, a contradiction. So p = q and P = Q or P = L by Step. If the latter, then by Lemma 2.8, we have L is cyclic of order p or 4 (if p = 2 ). Thus is supersolvable since / L is supersolvable by [9, 2.6]. Then we assume that P = Q. By hypothesis, all cyclic subgroups C of P of order p or (if p = 2 ) is weakly c -normal in. Then there exists a subnormal T such that = CT and

Int. J. Pure Appl. Sci. Technol., 5(2) (20), 9-30 28 C T C. We will deal with this from the following cases: Case : C =. Obviously T H, for otherwise, T H, then if T H =, C H and so = CT = HT, C T H T =, and C = H is a normal cyclic subgroup of. Thus is supersolvable,a contradiction. If T < H,then = CT = CH. Obviously C H, then = CH = H, a contradiction. Then we have H T. Then by step, T is supersolvable. But = CT is supersolvable by [4], a contradiction. Case 2: C T = C = C, where P is abelian or p is an odd prime. In this case, C T, then = CT = T is supersolvable by Step, a contradiction. Case 3: < C T = C C, where P is non-abelian and p = 2. By hypothesis and Lemma 2.7(4) C = 4,then C = 2. Since C is s -quasinormally embedded in, then there exists a s-quasinormal subgroup K of such that C is a Sylow 2-subgroup of K. Since for any Sylow q -subgroup Q of, where q 2, such that KQ = QK is a subgroup of. If KQ H, then by step, KQ is supersolvable, then K is normal in KQ and C is a Sylow 2-subgroup of KQ, and so C < KQ. We have CQ = QC and so C is s -quasinormal in. By Lemma p 2.5, O ( ) N ( C ) ( ). Thus C < so L = C and LQ = Q L. Then Q C L L, by [8, Hilfssatz C], is solvable, a contradiction. H Φ =. Step 4. ( ) If not, then there exists a prime divisor r of Φ ( ), and Let R be a Sylow r -subgroup of Φ ( ), then R is normal in. By Step 2, / R is supersolvable and so / ( ) Φ is supersolvable. Thus is supersolvable, a contradiction. Step 5. Conclusion. By Step 4 and [5, Lemma 2.6], then F ( H ) is the direct products of minimal normal subgroups of containing in H. By Step 2, F ( H ) = L. By Step and

Int. J. Pure Appl. Sci. Technol., 5(2) (20), 9-30 29 Lemma 2.7, exp ( L) = p or 4. By hypothesis, cyclic of L of order p or 4 (if p = 2 ) are weakly c -normal in. Thus by Lemma 2.8, L = x, for some x L \{ }, is a cyclic normal minimal subgroup of. Then by [9, 2.6], is supersolvable since / L is supersolvable by Step 2. The final contradiction completes the proof. Corollary 3.4 ( [, Theorem B])Let be a finite group. If there exists a normal subgroup H such that =H is supersoluble, ( H ) / H of H is pronormal in, then is supersoluble., 2 =, and every minimal subgroup Corollary 3.5 ([8, Theorem 3.]) Let H be a normal p -subgroup of such that / H is supersolvable. Suppose that every cyclic subgroup of H of order p or 4 (if p = 2 ) is π -quasinormal in, then is supersolvable. Acknowledgment The authors would like to thank the referee for the valuable suggestions and comments. The object is partially supported by the Scientific Research Fund of Sichuan Provincial Education Department (rant Number: 0ZA36) and the Scientific Research Fund of School of Science of SUSE(grant: 09LXYB02 and 09LXYA03), NSF of SUSE (rant Number: 200XJKYL07). References [] M. Asaad, On the supersolvability of finite groups I, Acta Mathematica Academiae Scientiarum Hungaricae, 38(-4) (98), 57-59. [2] M. Asaad, A.A. Heliel, On permutable subgroups of finite groups, Archiv der Mathematik, 80(2003), 3-8. [3] M. Asaad, M. Ramadan, and Y. Shaalan, Inuence of π -quasinormality on maximal subgroups of Sylow subgroups of Fitting subgroups of a finite group, Archiv der Mathematik, 56(99), 52-527. [4] R. Baer, Supersoluble immersion, Canadian Journal of Mathematics, (3) (959), 353-369. [5] A. Ballester-Bolinches, M.C. Pedraza-Aguilera, On minimal subgroups of finite groups, Acta Mathematica Hungarica, 73(4) (996), 335-342. [6] A. Ballester-Bolinches, M.C. Pedraza-Aguilera, Suficient conditions for supersolvability of finite groups, Journal of Pure and Applied Algebra, 27(998), 3-38. [7] W.E. Deskins, On quasinormal subgroups of finite groups, Mathematische Zeitschrift, 82(963), 25-32.

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