- Email: [email protected]
Finite Groups Whose Minimal Subgroups are Weakly -SUBGROUPS
Acta Mathematica Scientia 2012,32B(6):2295–2301 http://actams.wipm.ac.cn
FINITE GROUPS WHOSE MINIMAL SUBGROUPS ARE WEAKLY H-SUBGROUPS∗ M. M. Al-Mosa Al-Shomrani Department of Mathematics, Faculty of Science 80203, King Abdulaziz University, Jeddah 21589, Saudi Arabia
M. Ramadan Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
A. A. Heliel Department of Mathematics, Faculty of Science 80203, King Abdulaziz University, Jeddah 21589, Saudi Arabia Department of Mathematics, Beni-Suef University, Faculty of Science 62511, Beni-Suef, Egypt E-mail: [email protected]
Abstract Let G be a finite group. A subgroup H of G is called an H-subgroup in G if NG (H) ∩ H g ≤ H for all g ∈ G. A subgroup H of G is called a weakly H-subgroup in G if there exists a normal subgroup K of G such that G = HK and H ∩ K is an H-subgroup in G. In this paper, we investigate the structure of the finite group G under the assumption that every subgroup of G of prime order or of order 4 is a weakly H-subgroup in G. Our results improve and generalize several recent results in the literature. Key words c-normal subgroup; H-subgroup; p-nilpotent group; supersolvable group; generalized Fitting subgroup; saturated formation 2010 MR Subject Classification
1
20D10; 20D15; 20D20; 20F16
Introduction
All groups considered in this paper are finite. We use conventional notions and notation, as in Doerk and Hawkes [10]. It was well known that the normality of subgroups of a finite group greatly reflects the structure of the group. For example, Gasch¨ utz and It¨ o [12, Satz 5.7, p. 436] showed that a group G is solvable if all minimal subgroups of G are normal (a subgroup of prime order of G is called a minimal subgroup). Buckley [6] showed that a group G of odd order is supersolvable if all minimal subgroups are normal. In the recent years, various generalizations of normality were given, and many parallel results were obtained on such topics. For example, Wang [17] introduced the following concept: a subgroup H of a group G is said to be c-normal in G if G ∗ Received
June 9, 2011; revised March 21, 2012. The research supported by the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) represented by the Unit of Research Groups through the grant number (MG/31/01) for the group entitled “Abstract Algebra and its Applications”.
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has a normal subgroup K of G such that G = HK and H ∩ K ≤ HG , where HG = CoreG (H) is the largest normal subgroup of G contained in H. Afterwards there was a lot of research concerning this concept; see, for example [2, 7, 14–20]. Another generalization of normality was given by Bianchi et al. [8]. A subgroup H of a group G is said to be an H-subgroup in G if NG (H) ∩ H g ≤ H for all g ∈ G. They showed that a group G is a supersolvable T -group if and only if every subgroup of G is an H-subgroup (a T -group is a group in which every subnormal subgroup is normal). Later, Cs¨ org¨o and Herzog [9] showed that a group G is supersolvable if every cyclic subgroup of G of prime order or of order 4 is an H-subgroup. Some new conditions for a group to be p-nilpotent or supersolvable were given and many known results were generalized in Asaad [1] and in Guo and Wei [11] by using H-subgroup concept. One can easily find groups with H-subgroups that are not c-normal; for example, every Sylow subgroup of any nonabelian simple group G is an H-subgroup, but it is not c-normal in G. Conversely, there are also groups with c-normal subgroups that are not H-subgroups; for example, if G is the symmetric group of degree 4 and L = "(13)#, then L is c-normal but not Hsubgroup in G. In fact, about c-normality and H-subgroups, there is no inclusion-relationship between these two concepts. More recently, Asaad, Heliel and Al-Shomrani [3] introduced a new subgroup embedding property of a finite group, called a weakly H-subgroup, which is a generalization of both cnormality and H-subgroup. A subgroup H of a group G is said to be a weakly H-subgroup in G if there exists a normal subgroup K of G such that G = HK and H ∩ K ∈ H(G), where H(G) denotes the set of all H-subgroups of a group G. It is clear that each of c-normality and H-subgroup concepts implies weakly H-subgroup. The converse does not hold in general, see Examples 1, 2 and 3 in Asaad et al. [3]. The aim of this paper is to take the above mentioned studies further. More precisely, we investigate the structure of a finite group G under the assumption that every subgroup of G of prime order or of order 4 is a weakly H-subgroup in G. Our results unify and generalize many recent results that concerning c-normality and H-subgroup concepts in the literature.
2
Basic Definitions and Preliminaries
In this section, for the sake of convenience, we collect some definitions and state some known results from the literature which will be used in the sequel. Recall that a class of groups F is a formation provided that the following conditions are satisfied: (1) If G ∈ F, then G/N ∈ F, where N is any normal subgroup of G. (2) If G/M and G/N are both in F, then G/(M ∩N ) is also in F for any normal subgroups M and N of G. A formation F is said to be saturated if G/Φ(G) ∈ F implies that G belongs to F. Throughout this papere, U will denote the class of supersolvable groups. It is known that U is a saturated formation [12, Satz 8.6, p. 713]. A normal subgroup N of a group G is an F-hypercentral subgroup of G provided N possesses a chain of subgroups 1 = N0 N1 · · · Ns = N such that Ni+1 /Ni is an F-central chief factor of G (see [10, p. 387]). The product of all F-hypercentral subgroups of G is again
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an F-hypercentral subgroup, denoted by ZF (G), and called the F-hypercenter of G (see [10, IV 6.8]). For the formation U, the U-hypercenter of a group G will be denoted by ZU (G), that is, ZU (G) is the product of all normal subgroups N of G such that each chief factor of G below N has prime order. For more details about saturated formations (see [10, IV]). For any group G, the generalized Fitting subgroup F ∗ (G) is the set of all elements x of G which induce an inner automorphism on every chief factor of G. Clearly, F ∗ (G) is a characteristic subgroup of G. By [13, X 13], F ∗ (G) = 1 if G = 1. By [12, III 4.3], F (G) ≤ F ∗ (G), where F (G) denotes the Fitting subgroup of G. Lemma 2.1 Let P be a nontrivial normal p-subgroup of a group G, where p is an odd prime. If every minimal subgroup of P is normal in G, then P ≤ ZU (G). Proof Immediate consequence of [4, Theorem 1.1].
2
Lemma 2.2 (see [8, Theorem 6(2)]) Let G be a group and let H ∈ H(G). If H K ≤ G, then H K. Lemma 2.3 (see [21, Theorem 7.7, p. 32]) Let G be a group and N G, N ≤ ZU (G). Then ZU (G/N ) = ZU (G)/N . Lemma 2.4 (see [3, Lemma 2.2]) Let H be a subgroup of a group G. (1) If H is a weakly H-subgroup in G, H ≤ M ≤ G, then H is a weakly H-subgroup in M. (2) Let N G and N ≤ H. Then H is a weakly H-subgroup in G if and only if H/N is a weakly H-subgroup in G/N . (3) Let H be a p-subgroup of G and N a normal p -subgroup of G. If H is a weakly H-subgroup in G, then HN/N is a weakly H-subgroup in G/N . Lemma 2.5 (see [12, Satz 5.4, p. 434]) If G is a group which is not p-nilpotent but all of its proper subgroups are p-nilpotent, then it is a minimal non-nilpotent group (that is, G is not nilpotent but all of its proper subgroups are nilpotent). Lemma 2.6 Let P be a Sylow p-subgroup of a group G and suppose that one of the following conditions holds: (1) p = 2, and every subgroup of P of order 2 or of order 4 is normal in G. (2) p = 2, and every subgroup of order p of G lies in the center of G. Then G is p-nilpotent. Proof For (1), see [5, Lemma] and for (2), see [12, Satz 5.5, p. 435].
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Lemma 2.7 (see [21, Theorem 6.3, p. 221 and Corollary 7.8, p. 33]) Let P be a normal p-subgroup of a group G such that |G/CG (P )| is a power of p. Then P ≤ ZU (G). Lemma 2.8 (see [10, Propositin 3.11, p. 362]) If F1 and F2 are two saturated formations such that F1 ⊆F2 , then ZF1 (G)⊆ZF2 (G). Lemma 2.9 (see [13, X 13]) Let G be a group. Then (1) If F ∗ (G) is solvable, then F ∗ (G) = F (G). (2) CG (F ∗ (G)) ≤ F (G). Lemma 2.10 (see [10, Theorem 6.10, p. 390]) If F is a saturated formation, then [G , ZF (G)] = 1. F
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Results We first prove the following result:
Theorem 3.1 Let P be a nontrivial normal p-subgroup of a group G, where p is an odd prime. If every minimal subgroup of P is weakly H-subgroup in G, then P ≤ ZU (G). Proof We prove the theorem by induction on |G| + |P |. If every minimal subgroup of P is normal in G, then P ≤ ZU (G), by Lemma 2.1, and we are done. So, let us assume that P has a minimal subgroup H such that H is not normal in G. By hypothesis, H is a weakly H-subgroup in G. Then there exists a normal subgroup K of G such that G = HK and H ∩ K ∈ H(G). If H ∩ K = 1, then H ∩ K = H ∈ H(G) and, since H G, it follows that H G by Lemma 2.2, a contradiction. Thus, we may assume that H ∩ K = 1. Clearly, P = H(P ∩ K) and P ∩ K G. By hypothesis, every minimal subgroup of P ∩ K is a weakly H-subgroup in G. Hence, by induction on |G| + |P |, P ∩ K ≤ ZU (G). But P/(P ∩ K) is a normal subgroup of G/(P ∩ K) of order p, then P/(P ∩ K) ≤ ZU (G/(P ∩ K)) and since P ∩ K ≤ ZU (G), we have, by Lemma 2.3, that ZU (G/(P ∩ K)) = ZU (G)/(P ∩ K) and so P ≤ ZU (G). This completes the proof of the theorem. 2 We now prove: Theorem 3.2 Let P be a Sylow 2-subgroup of a group G. If every subgroup of P of order 2 or of order 4 is a weakly H-subgroup in G, then G is 2-nilpotent. Proof Assume that the result is false and let G be a counterexample of minimal order. By Lemma 2.4(1), the hypothesis is inherited by all proper subgroups of G. Hence G is a minimal non-p-nilpotent group. By Lemma 2.5, G is a minimal non-nilpotent group and so G = P Q, where P is a normal Sylow 2-subgroup and Q is a non-normal cyclic Sylow q-subgroup of G, q = 2. Moreover, the exponent of P is at most 4. If every subgroup of P of order 2 or of order 4 is normal in G, then, by Lemma 2.6(1), G is 2-nilpotent, a contradiction. Thus, there exists a subgroup H of P of order 2 or of order 4 such that H G. By hypothesis, H is a weakly H-subgroup in G. Then there exists a normal subgroup K of G such that G = HK and H ∩ K ∈ H(G). We may assume that K is a proper subgroup of G (for if K = G, then H ∈ H(G) and so, by Lemma 2.2, we have H G, a contradiction). Therefore, K is a normal nilpotent subgroup of G and this means that Q char K. Now, Q char K and K G imply that Q G, a final contradiction completing the proof of the theorem. 2 As an immediate consequence of Theorem 3.1 and Theorem 3.2, we have Corollary 3.3 Let P be a normal p-subgroup of a group G. If every subgroup of P of order p or of order 4 (if p = 2) is a weakly H-subgroup in G, then P ≤ ZU (G). Proof If p is an odd prime, then P ≤ ZU (G) by Theorem 3.1 and we are done. Thus we may assume that p = 2. Clearly, P Q is a subgroup of G, where Q is any Sylow q-subgroup of G, q = 2. By Lemma 2.4(1) and Theorem 3.2, P Q is 2-nilpotent and so P Q = P × Q. Hence |G/CG (P )| is a power of 2. Applying Lemma 2.7 yields P ≤ ZU (G). 2 We now prove: Theorem 3.4 Let p be the smallest prime dividing the order of a group G and let P be a Sylow p-subgroup of G. If every subgroup of order p or of order 4 (if p = 2) is a weakly H-subgroup in G, then G is p-nilpotent. Proof Assume that the result is false and let G be a counterexample of minimal order.
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So, by Theorem 3.2, p = 2. The hypothesis is inherited by all proper subgroups of G by Lemma 2.4(1). Therefore, G is a minimal non-p-nilpotent group and so G is a minimal non-nilpotent group. Then G = P Q, P G and Q is a non-normal cyclic Sylow q-subgroup of G, q = p. Let H be any minimal subgroup of P . By hypothesis, there exists a normal subgroup K of G such that G = HK and H ∩ K ∈ H(G). Clearly, H ∩ K G and so H ∩ K G by Lemma 2.2. If H G, then H ∩ K = 1 and therefore K is a proper subgroup of G and so K is nilpotent. Hence the Sylow q-subgroup of K is normal in G and therefore G is nilpotent, a contradiction. Thus, we may assume that every minimal subgroup of P must be normal in G and hence every minimal subgroup of P must be in the center of G. Applying Lemma 2.6(2) yields G is p-nilpotent, a contradiction completing the proof of the theorem. 2 Now we can prove: Theorem 3.5 Let F be a saturated formation containing the class of supersolvable groups U. A group G lies in F if and only if it has a normal subgroup H such that G/H ∈ F and every subgroup of H of prime order or of order 4 is a weakly H-subgroup in G. Proof We need only to prove the “ if ” part. We use induction on |G|. By using Lemma 2.4(1) and repeated applications of Theorem 3.4, H has a Sylow tower of supersolvable type. Then P H, where P is a Sylow p-subgroup of H and p is the largest prime dividing |H|. Since P char H and H G, we have P G. Clearly, H/P is a normal subgroup of G/P and (G/P )/(H/P ) ∼ = G/H ∈ F. Moreover, from Lemma 2.4, every subgroup of H/P of prime order or of order 4 is a weakly H-subgroup in G/P . Thus, by induction on |G|, we have G/P ∈ F. Since P ≤ ZU (G), by Corollary 3.3, and ZU (G) ≤ ZF (G), by Lemma 2.8, we have P ≤ ZF (G) and so G ∈ F. 2 The following corollaries are immediate consequences of Theorem 3.5. Corollary 3.6 Let G be a group with a normal subgroup H such that G/H is supersolvable. If every subgroup of H of prime order or of order 4 is a weakly H-subgroup in G, then G is supersolvable. Corollary 3.7 Let G be a group with a normal subgroup H such that (G/H) is nilpotent. If every subgroup of H of prime order or of order 4 is a weakly H-subgroup in G, then G is nilpotent. Corollary 3.8 If every subgroup of prime order or of order 4 of a group G is a weakly H-subgroup in G, then G is supersolvable. We now prove: Theorem 3.9 Let F be a saturated formation containing the class of supersolvable groups U. A group G lies in F if and only if it has a normal subgroup H such that G/H ∈ F and every subgroup of F ∗ (H) of prime order or of order 4 is a weakly H-subgroup in G. Proof We need only to prove the “ if ” part. By Lemma 2.4(1), we have that every subgroup of F ∗ (H) of prime order or of order 4 is a weakly H-subgroup in F ∗ (H). Corollary 3.8 implies that F ∗ (H) is supersolvable and hence, by Lemma 2.9(1), F ∗ (H) = F (H). By Corollary 3.3, F (H) ≤ ZU (G) and, since ZU (G) ≤ ZF (G), we have F (H) ≤ ZF (G). Hence G/CG (F (H)) ∈ F by Lemma 2.10. Since G/H and G/CG (F (H)) are in F, we have G/(H ∩ CG (F (H))) = G/CH (F (H)) ∈ F. But CH (F (H)) ≤ F (H) holds by Lemma 2.9(2) and the fact that F ∗ (H) = F (H). Then G/F (H) is an epimorphic image of G/CH (F (H)), thus G/F (H) ∈ F. Now, by applying Theorem 3.5, we get G ∈ F. 2
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As immediate consequence of Theorem 3.9, we have: Corollary 3.10 Let F be a saturated formation containing the class of supersolvable groups U. A group G lies in F if and only if it has a solvable normal subgroup H such that G/H ∈ F and every subgroup of F (H) of prime order or of order 4 is a weakly H-subgroup in G.
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Some Applications of the Results
In the paper one can find the following special cases of our main results: Corollary 4.1 (Asaad and Ezzat Mohamed [2]; see also Ramadan, Ezzat Mohamed and Heliel [16]) Let F be a saturated formation containing the class of supersolvable groups U. A group G lies in F if and only if it has a normal subgroup H such that G/H ∈ F and all subgroups of H of prime order or of order 4 are c-normal in G. Corollary 4.2 (Li and Guo [15]) Let G be a group with a normal subgroup H such that G/H is supersolvable. If all subgroups of H of prime order or of order 4 are c-normal in G, then G is supersolvable. Corollary 4.3 (Wei [19]; see also Yangming [22]) Let F be a saturated formation containing the class of supersolvable groups U. A group G lies in F if it has a solvable normal subgroup H such that G/H ∈ F and all subgroups of F (H) of prime order or of order 4 are c-normal in G. Corollary 4.4 (Wang [18]) Let G be a group with a solvable normal subgroup H such that G/H is supersolvable. If all subgroups of F (H) of prime order or of order 4 are c-normal in G, then G is supersolvable. Corollary 4.5 (Li and Guo [14]) Let G be a solvable group of odd order. If H is a normal subgroup of G such that G/H is supersolvable and all minimal subgroups which are contained in F (H) are c-normal in G, then G is supersolvable. Corollary 4.6 (Wei, Wang and Li [20]) Let F be a saturated formation containing the class of supersolvable groups U. Assume that G is a group with a normal subgroup H such that G/H ∈ F. If all minimal subgroups and all cyclic subgroups of F ∗ (H) are c-normal in G, then G ∈ F. Corollary 4.7 (Wang [18]) Let G be a group with a normal subgroup H such that G/H is supersolvable. If all subgroups of F ∗ (H) of prime order or of order 4 are c-normal in G, then G is supersolvable. References [1] Asaad M. On p-nilpotence and supersolvability of finite groups. Comm Algebra, 2006, 34: 189–195 [2] Asaad M, Ezzat Mohamed M. On c-normality of finite groups. J Aust Math Soc, 2005, 78: 297–304 [3] Asaad M, Heliel A A, Al-Mosa Al-Shomrani M M. On weakly H-subgroups of finite groups. Comm Algebra, 2012, DOI: 10.1080/00927872.2011.591218 [4] Asaad M, Ramadan M. Finite groups whose minimal subgroups are c-supplemented. Comm Algebra, 2008, 36: 1034–1040 [5] Asaad M, Li S. On minimal subgroups of finite groups II. Comm Algebra, 1996, 24(14): 4603–4606 [6] Buckley J. Finite groups whose minimal subgroups are normal. Math Z, 1970, 116: 15–17 [7] Ballester-Bolinches A, Wang Y. Finite groups with some c-normal minimal subgroups. J Pure Appl Algebra, 2000, 153: 121–127
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[8] Bianchi M, Gillio Berta Mauri A, Herzog M, Verardi L. On finite solvable groups in which normality is a transitive relation. J Group Theory, 2000, 3: 147–156 [9] Cs¨ org¨ o P, Herzog M. On supersolvable groups and the nilpotator. Comm Algebra, 2004, 32: 609–620 [10] Doerk K, Hawkes T. Finite Soluble Groups. Berlin, New York: Walter de Gruyter, 1992 [11] Guo X, Wei X. The influence of H-subgroups on the structure of finite groups. J Group Theory, 2010, 13: 267–276 [12] Huppert B. Endliche Gruppen I. Berlin, Heidelberg, New York: Springer-Verlage, 1979 [13] Huppert B, Blackburn N. Finite Groups III. Berlin, Heidelberg, New York: Springer-Verlage, 1982 [14] Li D, Guo X. The influence of c-normality of subgroups on the structure of finite groups II. Comm Algebra, 1998, 26(6): 1913–1922 [15] Li D, Guo X. The influence of c-normality of subgroups on the structure of finite groups. J Pure Appl Algebra, 2000, 150: 53–60 [16] Ramadan M, Ezzat Mohamed M, Heliel A A. On c-normality of certain subgroups of prime power order of finite groups. Arch Math, 2005, 85: 203–210 [17] Wang Y. C-normality of groups and its properties. J Algebra, 1996, 180: 954–965 [18] Wang Y. The influence of minimal subgroups on the structure of finite groups. Acta Math Sin, 2000, 16(1): 63–70 [19] Wei H Q. On c-normal maximal and minimal subgroups of Sylow subgroups of finite groups. Comm Algebra, 2001, 29(5): 2193–2200 [20] Wei H Q, Wang Y M, Li Y M. On c-normal maximal and minimal subgroups of Sylow subgroups of finite groups II. Comm Algebra, 2003, 31(10): 4807–4816 [21] Weinstein M (editor). Between Nilpotent and Solvable. Passaic: Polygonal Publishing House, 1982 [22] Li Y M. Some notes on the minimal subgroups of Fitting subgroups of finite groups. J Pure Appl Algebra, 2002, 171: 289–294
FINITE GROUPS WHOSE MINIMAL SUBGROUPS ARE WEAKLY H-SUBGROUPS∗ M. M. Al-Mosa Al-Shomrani Department of Mathematics, Faculty of Science 80203, King Abdulaziz University, Jeddah 21589, Saudi Arabia
M. Ramadan Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
A. A. Heliel Department of Mathematics, Faculty of Science 80203, King Abdulaziz University, Jeddah 21589, Saudi Arabia Department of Mathematics, Beni-Suef University, Faculty of Science 62511, Beni-Suef, Egypt E-mail: [email protected]
Abstract Let G be a finite group. A subgroup H of G is called an H-subgroup in G if NG (H) ∩ H g ≤ H for all g ∈ G. A subgroup H of G is called a weakly H-subgroup in G if there exists a normal subgroup K of G such that G = HK and H ∩ K is an H-subgroup in G. In this paper, we investigate the structure of the finite group G under the assumption that every subgroup of G of prime order or of order 4 is a weakly H-subgroup in G. Our results improve and generalize several recent results in the literature. Key words c-normal subgroup; H-subgroup; p-nilpotent group; supersolvable group; generalized Fitting subgroup; saturated formation 2010 MR Subject Classification
1
20D10; 20D15; 20D20; 20F16
Introduction
All groups considered in this paper are finite. We use conventional notions and notation, as in Doerk and Hawkes [10]. It was well known that the normality of subgroups of a finite group greatly reflects the structure of the group. For example, Gasch¨ utz and It¨ o [12, Satz 5.7, p. 436] showed that a group G is solvable if all minimal subgroups of G are normal (a subgroup of prime order of G is called a minimal subgroup). Buckley [6] showed that a group G of odd order is supersolvable if all minimal subgroups are normal. In the recent years, various generalizations of normality were given, and many parallel results were obtained on such topics. For example, Wang [17] introduced the following concept: a subgroup H of a group G is said to be c-normal in G if G ∗ Received
June 9, 2011; revised March 21, 2012. The research supported by the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) represented by the Unit of Research Groups through the grant number (MG/31/01) for the group entitled “Abstract Algebra and its Applications”.
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has a normal subgroup K of G such that G = HK and H ∩ K ≤ HG , where HG = CoreG (H) is the largest normal subgroup of G contained in H. Afterwards there was a lot of research concerning this concept; see, for example [2, 7, 14–20]. Another generalization of normality was given by Bianchi et al. [8]. A subgroup H of a group G is said to be an H-subgroup in G if NG (H) ∩ H g ≤ H for all g ∈ G. They showed that a group G is a supersolvable T -group if and only if every subgroup of G is an H-subgroup (a T -group is a group in which every subnormal subgroup is normal). Later, Cs¨ org¨o and Herzog [9] showed that a group G is supersolvable if every cyclic subgroup of G of prime order or of order 4 is an H-subgroup. Some new conditions for a group to be p-nilpotent or supersolvable were given and many known results were generalized in Asaad [1] and in Guo and Wei [11] by using H-subgroup concept. One can easily find groups with H-subgroups that are not c-normal; for example, every Sylow subgroup of any nonabelian simple group G is an H-subgroup, but it is not c-normal in G. Conversely, there are also groups with c-normal subgroups that are not H-subgroups; for example, if G is the symmetric group of degree 4 and L = "(13)#, then L is c-normal but not Hsubgroup in G. In fact, about c-normality and H-subgroups, there is no inclusion-relationship between these two concepts. More recently, Asaad, Heliel and Al-Shomrani [3] introduced a new subgroup embedding property of a finite group, called a weakly H-subgroup, which is a generalization of both cnormality and H-subgroup. A subgroup H of a group G is said to be a weakly H-subgroup in G if there exists a normal subgroup K of G such that G = HK and H ∩ K ∈ H(G), where H(G) denotes the set of all H-subgroups of a group G. It is clear that each of c-normality and H-subgroup concepts implies weakly H-subgroup. The converse does not hold in general, see Examples 1, 2 and 3 in Asaad et al. [3]. The aim of this paper is to take the above mentioned studies further. More precisely, we investigate the structure of a finite group G under the assumption that every subgroup of G of prime order or of order 4 is a weakly H-subgroup in G. Our results unify and generalize many recent results that concerning c-normality and H-subgroup concepts in the literature.
2
Basic Definitions and Preliminaries
In this section, for the sake of convenience, we collect some definitions and state some known results from the literature which will be used in the sequel. Recall that a class of groups F is a formation provided that the following conditions are satisfied: (1) If G ∈ F, then G/N ∈ F, where N is any normal subgroup of G. (2) If G/M and G/N are both in F, then G/(M ∩N ) is also in F for any normal subgroups M and N of G. A formation F is said to be saturated if G/Φ(G) ∈ F implies that G belongs to F. Throughout this papere, U will denote the class of supersolvable groups. It is known that U is a saturated formation [12, Satz 8.6, p. 713]. A normal subgroup N of a group G is an F-hypercentral subgroup of G provided N possesses a chain of subgroups 1 = N0 N1 · · · Ns = N such that Ni+1 /Ni is an F-central chief factor of G (see [10, p. 387]). The product of all F-hypercentral subgroups of G is again
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an F-hypercentral subgroup, denoted by ZF (G), and called the F-hypercenter of G (see [10, IV 6.8]). For the formation U, the U-hypercenter of a group G will be denoted by ZU (G), that is, ZU (G) is the product of all normal subgroups N of G such that each chief factor of G below N has prime order. For more details about saturated formations (see [10, IV]). For any group G, the generalized Fitting subgroup F ∗ (G) is the set of all elements x of G which induce an inner automorphism on every chief factor of G. Clearly, F ∗ (G) is a characteristic subgroup of G. By [13, X 13], F ∗ (G) = 1 if G = 1. By [12, III 4.3], F (G) ≤ F ∗ (G), where F (G) denotes the Fitting subgroup of G. Lemma 2.1 Let P be a nontrivial normal p-subgroup of a group G, where p is an odd prime. If every minimal subgroup of P is normal in G, then P ≤ ZU (G). Proof Immediate consequence of [4, Theorem 1.1].
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Lemma 2.2 (see [8, Theorem 6(2)]) Let G be a group and let H ∈ H(G). If H K ≤ G, then H K. Lemma 2.3 (see [21, Theorem 7.7, p. 32]) Let G be a group and N G, N ≤ ZU (G). Then ZU (G/N ) = ZU (G)/N . Lemma 2.4 (see [3, Lemma 2.2]) Let H be a subgroup of a group G. (1) If H is a weakly H-subgroup in G, H ≤ M ≤ G, then H is a weakly H-subgroup in M. (2) Let N G and N ≤ H. Then H is a weakly H-subgroup in G if and only if H/N is a weakly H-subgroup in G/N . (3) Let H be a p-subgroup of G and N a normal p -subgroup of G. If H is a weakly H-subgroup in G, then HN/N is a weakly H-subgroup in G/N . Lemma 2.5 (see [12, Satz 5.4, p. 434]) If G is a group which is not p-nilpotent but all of its proper subgroups are p-nilpotent, then it is a minimal non-nilpotent group (that is, G is not nilpotent but all of its proper subgroups are nilpotent). Lemma 2.6 Let P be a Sylow p-subgroup of a group G and suppose that one of the following conditions holds: (1) p = 2, and every subgroup of P of order 2 or of order 4 is normal in G. (2) p = 2, and every subgroup of order p of G lies in the center of G. Then G is p-nilpotent. Proof For (1), see [5, Lemma] and for (2), see [12, Satz 5.5, p. 435].
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Lemma 2.7 (see [21, Theorem 6.3, p. 221 and Corollary 7.8, p. 33]) Let P be a normal p-subgroup of a group G such that |G/CG (P )| is a power of p. Then P ≤ ZU (G). Lemma 2.8 (see [10, Propositin 3.11, p. 362]) If F1 and F2 are two saturated formations such that F1 ⊆F2 , then ZF1 (G)⊆ZF2 (G). Lemma 2.9 (see [13, X 13]) Let G be a group. Then (1) If F ∗ (G) is solvable, then F ∗ (G) = F (G). (2) CG (F ∗ (G)) ≤ F (G). Lemma 2.10 (see [10, Theorem 6.10, p. 390]) If F is a saturated formation, then [G , ZF (G)] = 1. F
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Results We first prove the following result:
Theorem 3.1 Let P be a nontrivial normal p-subgroup of a group G, where p is an odd prime. If every minimal subgroup of P is weakly H-subgroup in G, then P ≤ ZU (G). Proof We prove the theorem by induction on |G| + |P |. If every minimal subgroup of P is normal in G, then P ≤ ZU (G), by Lemma 2.1, and we are done. So, let us assume that P has a minimal subgroup H such that H is not normal in G. By hypothesis, H is a weakly H-subgroup in G. Then there exists a normal subgroup K of G such that G = HK and H ∩ K ∈ H(G). If H ∩ K = 1, then H ∩ K = H ∈ H(G) and, since H G, it follows that H G by Lemma 2.2, a contradiction. Thus, we may assume that H ∩ K = 1. Clearly, P = H(P ∩ K) and P ∩ K G. By hypothesis, every minimal subgroup of P ∩ K is a weakly H-subgroup in G. Hence, by induction on |G| + |P |, P ∩ K ≤ ZU (G). But P/(P ∩ K) is a normal subgroup of G/(P ∩ K) of order p, then P/(P ∩ K) ≤ ZU (G/(P ∩ K)) and since P ∩ K ≤ ZU (G), we have, by Lemma 2.3, that ZU (G/(P ∩ K)) = ZU (G)/(P ∩ K) and so P ≤ ZU (G). This completes the proof of the theorem. 2 We now prove: Theorem 3.2 Let P be a Sylow 2-subgroup of a group G. If every subgroup of P of order 2 or of order 4 is a weakly H-subgroup in G, then G is 2-nilpotent. Proof Assume that the result is false and let G be a counterexample of minimal order. By Lemma 2.4(1), the hypothesis is inherited by all proper subgroups of G. Hence G is a minimal non-p-nilpotent group. By Lemma 2.5, G is a minimal non-nilpotent group and so G = P Q, where P is a normal Sylow 2-subgroup and Q is a non-normal cyclic Sylow q-subgroup of G, q = 2. Moreover, the exponent of P is at most 4. If every subgroup of P of order 2 or of order 4 is normal in G, then, by Lemma 2.6(1), G is 2-nilpotent, a contradiction. Thus, there exists a subgroup H of P of order 2 or of order 4 such that H G. By hypothesis, H is a weakly H-subgroup in G. Then there exists a normal subgroup K of G such that G = HK and H ∩ K ∈ H(G). We may assume that K is a proper subgroup of G (for if K = G, then H ∈ H(G) and so, by Lemma 2.2, we have H G, a contradiction). Therefore, K is a normal nilpotent subgroup of G and this means that Q char K. Now, Q char K and K G imply that Q G, a final contradiction completing the proof of the theorem. 2 As an immediate consequence of Theorem 3.1 and Theorem 3.2, we have Corollary 3.3 Let P be a normal p-subgroup of a group G. If every subgroup of P of order p or of order 4 (if p = 2) is a weakly H-subgroup in G, then P ≤ ZU (G). Proof If p is an odd prime, then P ≤ ZU (G) by Theorem 3.1 and we are done. Thus we may assume that p = 2. Clearly, P Q is a subgroup of G, where Q is any Sylow q-subgroup of G, q = 2. By Lemma 2.4(1) and Theorem 3.2, P Q is 2-nilpotent and so P Q = P × Q. Hence |G/CG (P )| is a power of 2. Applying Lemma 2.7 yields P ≤ ZU (G). 2 We now prove: Theorem 3.4 Let p be the smallest prime dividing the order of a group G and let P be a Sylow p-subgroup of G. If every subgroup of order p or of order 4 (if p = 2) is a weakly H-subgroup in G, then G is p-nilpotent. Proof Assume that the result is false and let G be a counterexample of minimal order.
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So, by Theorem 3.2, p = 2. The hypothesis is inherited by all proper subgroups of G by Lemma 2.4(1). Therefore, G is a minimal non-p-nilpotent group and so G is a minimal non-nilpotent group. Then G = P Q, P G and Q is a non-normal cyclic Sylow q-subgroup of G, q = p. Let H be any minimal subgroup of P . By hypothesis, there exists a normal subgroup K of G such that G = HK and H ∩ K ∈ H(G). Clearly, H ∩ K G and so H ∩ K G by Lemma 2.2. If H G, then H ∩ K = 1 and therefore K is a proper subgroup of G and so K is nilpotent. Hence the Sylow q-subgroup of K is normal in G and therefore G is nilpotent, a contradiction. Thus, we may assume that every minimal subgroup of P must be normal in G and hence every minimal subgroup of P must be in the center of G. Applying Lemma 2.6(2) yields G is p-nilpotent, a contradiction completing the proof of the theorem. 2 Now we can prove: Theorem 3.5 Let F be a saturated formation containing the class of supersolvable groups U. A group G lies in F if and only if it has a normal subgroup H such that G/H ∈ F and every subgroup of H of prime order or of order 4 is a weakly H-subgroup in G. Proof We need only to prove the “ if ” part. We use induction on |G|. By using Lemma 2.4(1) and repeated applications of Theorem 3.4, H has a Sylow tower of supersolvable type. Then P H, where P is a Sylow p-subgroup of H and p is the largest prime dividing |H|. Since P char H and H G, we have P G. Clearly, H/P is a normal subgroup of G/P and (G/P )/(H/P ) ∼ = G/H ∈ F. Moreover, from Lemma 2.4, every subgroup of H/P of prime order or of order 4 is a weakly H-subgroup in G/P . Thus, by induction on |G|, we have G/P ∈ F. Since P ≤ ZU (G), by Corollary 3.3, and ZU (G) ≤ ZF (G), by Lemma 2.8, we have P ≤ ZF (G) and so G ∈ F. 2 The following corollaries are immediate consequences of Theorem 3.5. Corollary 3.6 Let G be a group with a normal subgroup H such that G/H is supersolvable. If every subgroup of H of prime order or of order 4 is a weakly H-subgroup in G, then G is supersolvable. Corollary 3.7 Let G be a group with a normal subgroup H such that (G/H) is nilpotent. If every subgroup of H of prime order or of order 4 is a weakly H-subgroup in G, then G is nilpotent. Corollary 3.8 If every subgroup of prime order or of order 4 of a group G is a weakly H-subgroup in G, then G is supersolvable. We now prove: Theorem 3.9 Let F be a saturated formation containing the class of supersolvable groups U. A group G lies in F if and only if it has a normal subgroup H such that G/H ∈ F and every subgroup of F ∗ (H) of prime order or of order 4 is a weakly H-subgroup in G. Proof We need only to prove the “ if ” part. By Lemma 2.4(1), we have that every subgroup of F ∗ (H) of prime order or of order 4 is a weakly H-subgroup in F ∗ (H). Corollary 3.8 implies that F ∗ (H) is supersolvable and hence, by Lemma 2.9(1), F ∗ (H) = F (H). By Corollary 3.3, F (H) ≤ ZU (G) and, since ZU (G) ≤ ZF (G), we have F (H) ≤ ZF (G). Hence G/CG (F (H)) ∈ F by Lemma 2.10. Since G/H and G/CG (F (H)) are in F, we have G/(H ∩ CG (F (H))) = G/CH (F (H)) ∈ F. But CH (F (H)) ≤ F (H) holds by Lemma 2.9(2) and the fact that F ∗ (H) = F (H). Then G/F (H) is an epimorphic image of G/CH (F (H)), thus G/F (H) ∈ F. Now, by applying Theorem 3.5, we get G ∈ F. 2
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As immediate consequence of Theorem 3.9, we have: Corollary 3.10 Let F be a saturated formation containing the class of supersolvable groups U. A group G lies in F if and only if it has a solvable normal subgroup H such that G/H ∈ F and every subgroup of F (H) of prime order or of order 4 is a weakly H-subgroup in G.
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Some Applications of the Results
In the paper one can find the following special cases of our main results: Corollary 4.1 (Asaad and Ezzat Mohamed [2]; see also Ramadan, Ezzat Mohamed and Heliel [16]) Let F be a saturated formation containing the class of supersolvable groups U. A group G lies in F if and only if it has a normal subgroup H such that G/H ∈ F and all subgroups of H of prime order or of order 4 are c-normal in G. Corollary 4.2 (Li and Guo [15]) Let G be a group with a normal subgroup H such that G/H is supersolvable. If all subgroups of H of prime order or of order 4 are c-normal in G, then G is supersolvable. Corollary 4.3 (Wei [19]; see also Yangming [22]) Let F be a saturated formation containing the class of supersolvable groups U. A group G lies in F if it has a solvable normal subgroup H such that G/H ∈ F and all subgroups of F (H) of prime order or of order 4 are c-normal in G. Corollary 4.4 (Wang [18]) Let G be a group with a solvable normal subgroup H such that G/H is supersolvable. If all subgroups of F (H) of prime order or of order 4 are c-normal in G, then G is supersolvable. Corollary 4.5 (Li and Guo [14]) Let G be a solvable group of odd order. If H is a normal subgroup of G such that G/H is supersolvable and all minimal subgroups which are contained in F (H) are c-normal in G, then G is supersolvable. Corollary 4.6 (Wei, Wang and Li [20]) Let F be a saturated formation containing the class of supersolvable groups U. Assume that G is a group with a normal subgroup H such that G/H ∈ F. If all minimal subgroups and all cyclic subgroups of F ∗ (H) are c-normal in G, then G ∈ F. Corollary 4.7 (Wang [18]) Let G be a group with a normal subgroup H such that G/H is supersolvable. If all subgroups of F ∗ (H) of prime order or of order 4 are c-normal in G, then G is supersolvable. References [1] Asaad M. On p-nilpotence and supersolvability of finite groups. Comm Algebra, 2006, 34: 189–195 [2] Asaad M, Ezzat Mohamed M. On c-normality of finite groups. J Aust Math Soc, 2005, 78: 297–304 [3] Asaad M, Heliel A A, Al-Mosa Al-Shomrani M M. On weakly H-subgroups of finite groups. Comm Algebra, 2012, DOI: 10.1080/00927872.2011.591218 [4] Asaad M, Ramadan M. Finite groups whose minimal subgroups are c-supplemented. Comm Algebra, 2008, 36: 1034–1040 [5] Asaad M, Li S. On minimal subgroups of finite groups II. Comm Algebra, 1996, 24(14): 4603–4606 [6] Buckley J. Finite groups whose minimal subgroups are normal. Math Z, 1970, 116: 15–17 [7] Ballester-Bolinches A, Wang Y. Finite groups with some c-normal minimal subgroups. J Pure Appl Algebra, 2000, 153: 121–127
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