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On 𝔍hq-supplemented Subgroups of a Finite Group



M. Ezzat Mohamed, Mohammed M. Al-Shomrani and M.I. Elashiry
 
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ABSTRACT

Background and Objective: A subgroup H of a finite group G is quasinormal in G if it permutes with every subgroup of G. A subgroup H of a finite group G is 𝔍hp-supplemented in G if G has a quasinormal subgroup N such that HN is a Hall subgroup of N and (H∩N)HG/HG<Z𝔍(G/HG), where HG is the core of H in G and Z𝔍(G/HG) is the hypercenter of G/HG. The main objective of this study is to study the structure of a finite group under the assumption that some subgroups of prime power order are 𝔍hp-supplemented in the group. Methodology: This study can improve previous results by studying the properties of the concept of 𝔍hq-supplemented and using some lemmas on these concept. Results: Results clearly reveal the influence the concept of 𝔍hq-supplemented of some subgroups of prime power order on the group. Conclusion: This study improves and extends some results of super solvability of the group by using the concept of 𝔍hq-supplemented.

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M. Ezzat Mohamed, Mohammed M. Al-Shomrani and M.I. Elashiry, 2017. On 𝔍hq-supplemented Subgroups of a Finite Group. Journal of Applied Sciences, 17: 148-152.

DOI: 10.3923/jas.2017.148.152

URL: https://scialert.net/abstract/?doi=jas.2017.148.152
 
Received: November 06, 2016; Accepted: December 22, 2016; Published: February 15, 2017


Copyright: © 2017. This is an open access article distributed under the terms of the creative commons attribution License, which permits unrestricted use, distribution and reproduction in any medium, provided the original author and source are credited.

INTRODUCTION

All groups considered in this study will be finite and G always means a finite group. The conventional notions and notations, as in Doerk and Hawkes1.

Recall that a formation is a hypomorph 𝔍 of groups such that each group G has the smallest normal subgroup whose quotient is still in 𝔍. A formation 𝔍 is said to be saturated if it contains each group G with G/Φ(G)∈𝔍. In this study, the symbol U denote the class of supersolvable groups. Clearly, U is a saturated formation. A formation 𝔍 is said to be S-closed (Sn-closed) if it contains every subgroup (every normal subgroup, respectively) of all its groups. Let [A]B stand for the semi-product of two groups A and B. For a class 𝔍 of groups, a chief factor H/K of a group G is called 𝔍-central2 if [H/K](G/CG (H/K))∈𝔍. The symbol Z𝔍(G) denotes the 𝔍-heypercenter of a group G, that is the product of all such H of G whose G-chief factors are 𝔍-central.

Recall that two subgroups H and K of a group G are said to permute if HK = KH. A subgroup H of a group G is called quasinormal (or permutable) in G if it permutes with all subgroups of G. A subgroup H of a group G is said to be c-normal in G3 if G has a normal subgroup N such that G = HN and H∩N<HG, where HG = CoreG (H) = ∩Hg is the maximal normal subgroup of G which is contained in H. Guo et al.4 introduced the following concept. They defined that the subgroup H of a group G is said to be 𝔍h-normal if there exists a normal subgroup K of G such that HK is a normal Hall subgroup of G and (H∩K)HG/HG<Z𝔍 (G/HG), the researchers have obtained some interesting results5. In spite of the fact that the c-normal and 𝔍h-normal are quite different generalizations of normality there are several analogous results which were obtained independently for c-normal and 𝔍h-normal subgroups. Recently, Mohamed et al.6, introduced the following concept which covers normality, c-normality and 𝔍h-normality.

Definition: A subgroup H of G is 𝔍hq-supplemented in G if G has a quasinormal subgroup N such that HN is a Hall subgroup of G and (H∩K)HG/HG<Z𝔍 (G/HGG).

Several studies investigated the relationship between the properties of subgroups of a finite group G and the structure of G7-10. Specially, maximal subgroups of sylow subgroups play an important role in determining the structure of a finite group. They have been studied by many scholars. A typical result in this direction is due to Srinivasan11. It states that a group G is supersolvable if it has a normal subgroup N with supersolvable quotient G/N such that the maximal subgroups of the sylow subgroups of N are normal in G.

The main goal of this study is to report the structure of G under assumption that the maximal subgroups of the sylow subgroups of G are Uhq-supplemented in G and to discuss some applications.

Preliminaries
Lemma 2.1:
Let G be a group and H<K<G. Then:

(a) H is 𝔍hq-supplemented in G if and only if G has a quasinormal subgroup N such that HN is a Hall subgroup of G, HG<N and (H/HG)∩(N/HG)<Z𝔍 (G/HG)
(b) If H is a normal subgroup of G and K is 𝔍hq-supplemented in G, then K/H is 𝔍hq-supplemented in G/H
(c) If H is a normal subgroup of G, then the subgroup EH/H is 𝔍hq-supplemented in G/H for every 𝔍hq-supplemented in G subgroup E satisfying (∣H∣, ∣E∣) = 1
(d) If H is 𝔍hq-supplemented in G and 𝔍 is S-closed, then H is 𝔍hq-supplemented in K

Proof: Guo2

Lemma 2.2: If pn is the smallest prime dividing the order of a group G and p1 is the largest prime dividing the order of G, where pn≠p1, then G possesses supersolvable subgroups H and K with ∣G:H∣ = pn and ∣G:K∣ = p1 if and only if G is supersolvable.

Proof: Ramadan et al.14

RESULTS

Lemma 3.1: Let p be the smallest prime dividing the order of G and let Gp be a sylow p-subgroup of G. If the maximal subgroups of Gp are Uhq-supplemented in G, then G is p-nilpotent.

Proof: Suppose the result is false and let G be a counter-example of minimal order. For the sake of clarity, the proof breaks into four parts:

(1) OP(G) = 1

Suppose OP(G)≠1. Now consider the group G/Op(G). Clearly GpOp(G)/Op(G) is a sylow p-subgroup of G/Op(G). Let POp(G)/Op(G) be a maximal subgroup of GpOp(G)/Op(G). Then P is a maximal subgroup of Gp. By hypothesis, P is Uhq-supplemented in G. So POp(G)/Op(G)is Uhq-supplemented in G/Op(G), by Lemma 2.1, then the hypothesis of theorem hold on G/Op(G). Hence, G/Op(G) is p-nilpotent by the minimality of G and so does G; a contradiction.

(2) ZU (G) = 1

Suppose ZU (G)≠1. If ZU (G) is not p-subgroup of G, then ZU (G) has a normal sylow q-subgroup Q such that q is the largest prime dividing the order of ZU (G), as ZU (G) is supersolvable. Clearly q≠p. Since Q characteristic in ZU (G) and ZU (G) is a normal subgroup of G, it follows that Q is a normal subgroup of G. Then 1≠Q<Op(G), a contradiction with 1. Now, it follows that ZU (G) is a p-subgroup of G, hence there exists a normal subgroup N of G contained in ZU (G) such that ∣N∣ = p. Consider the group G/N. Clearly Gp/N be a sylow p-subgroup of G/N. By hypothesis and Lemma 2.1 (b), the maximal subgroups of Gp/N are Uhq-supplemented in G/N. Now, it follows that G/N is p-nilpotent by the minimality of G, then G/N contains a normal p’-Hall subgroup K/N and since N is a cyclic subgroup of order p, it follows by Huppert13, that K is p-nilpotent and also does G; a contradiction.

(3) Op(G)≠1

Suppose Op(G) = 1. Then HG = 1, for all subgroups H of Gp. Let P be a maximal subgroup of Gp. By hypothesis, P is Uhq-supplemented in G, then by Lemma 2.1 (a), there exists a quasinormal subgroup N of G such that PN is a Hall subgroup of G, PG<N and P/PG∩N/PG<ZU (G/PG). Since PG = 1, it follows that P∩N<ZU (G) and since ZU(G) = 1 by 2, it follows that P∩N = 1. Since PN is a Hall subgroup of G, it follows that P<Gp<PN and so Gp = P(Gp∩N). Now, it follows that ∣Gp∩N∣ = ∣Gp:P∣ = p and so Gp∩N is a cyclic sylow p-subgroup of N, then N is p-nilpotent by Huppert13. Thus, there exists a normal p’-Hall subgroup H of N. Since N is quasinormal subgroup of G, it follows that N is subnormal subgroup of G. So, H is also subnormal subgroup of G. Since PN is a Hall subgroup of G and H is a p’-Hall subgroup of N, it follows that H is a p’-Hall subgroup of G, i.e., H is a subnormal p’-Hall subgroup of G. Now, it follows that H is a normal p’-Hall subgroup of G, then H = 1, as OP(G) = 1 from 1. Thus N = Gp∩N is quasinormal subgroup of G of order p and so 1≠N<ZU(G); a contradiction with 2.

Final contradiction: Let H be a minimal normal subgroup of G contained in OP(G), then H≠1 as OP(G)≠1 by 3. Clearly the hypothesis of the theorem can be hold on the group G/H, by Lemma 2.1 (b), then G/H is p-nilpotent by the minimality of G. Since the class of all p-nilpotent groups is a formation, it follows that H is a unique minimal normal subgroup of G contained in OP(G). If H<Φ(Gp)1, H<Φ(G) and since G/H is p-nilpotent, it follows that G/Φ (G) is p-nilpotent. Now, it follows that G is also p-nilpotent as the class of all p-nilpotent groups is a saturated formation; a contradiction, then H is not subgroup of Φ(Gp). So, there exists a maximal subgroup P of Gp such that H is not subgroup of P. Clearly Gp = PH. By hypothesis P is Uhq-supplemented in G, then by Lemma 2.1 (a), there exists a quasinormal subgroup N of G such that PN is a Hall subgroup of G, PG<N and P/PG∩N/PG<ZU(G/PG). Since H is a unique minimal normal subgroup of G contained in OP(G), it follows that H<PG<P; a contradiction. Thus PG = 1, then it follows that P∩N<ZU(G). But ZU (G) = 1 by 2. Thus, P∩N = 1, then it follows that GP = P(GP∈N) and ∣GP∩N∣ = ∣GP:P∣ = p. By repeated the proof of 3, it follows that 1≠N<ZU(G); a final contradiction with 2. As an immediate consequence of Theorem 3.1.

Corollary 3.2: If the maximal subgroups of the sylow subgroups of a group G are Uhq-supplemented in G except for the largest prime dividing the order of G, then G possesses an ordered sylow tower.

Proof: By Theorem 3.1, G is p-nilpotent, where p is the smallest prime dividing the order of G, then G = GpK, where Gp is a sylow P-subgroup of G and K is a normal p’-Hall subgroup of G. By Lemma 2.1 (d), the hypothesis carries over K. Then K possesses an ordered sylow tower by the induction on the order of G, therefore; G possesses an ordered sylow tower.

Now prove that:

Theorem 3.3: Let P be a normal p-subgroup of a group G such that G/P∈U. If the maximal subgroups of P are Uhq-supplemented in G, then G∈U.

Proof: Suppose the result is false and let G be a counter-example of minimal order. Let Gp be a sylow p-subgroup of G. Treatment can be done by two cases:

Case 1: P = Gp

Then by Schur Zassenhaus’s Theorem, G/Gp≅K∈U, where K is a p’-Hall subgroup of G. Let N be a minimal normal subgroup of G contained in Gp. Then (G/N)/(Gp/N)≅G/Gp∈U. By hypothesis and Lemma 2.1 (b), the maximal subgroups of G/N are Uhq-supplemented in G/N. Then G/N∈U, by the minimality of G. Since the class of all supersolvable groups is a saturated formation, it follows that N is a unique minimal normal subgroup of G contained in Gp. If Φ (Gp)≠1, then N<Φ(Gp) and so N<Φ(G) by Doerk and Hawkes1. Clearly G/Φ(G)≅(G/N)/(Φ(G)/N)∈U. Then G∈U, as the class U is a saturated formation; a contradiction. Thus Φ(Gp). Now, it follows that there exists a maximal subgroup P1 of Gp such that N is not subgroup of P1. By hypothesis P1 is Uhq-supplemented in G, then by Lemma 2.1 (a), there exists a quasinormal subgroup H of G, (P1)G<h and (P1/(P1)G)∩(H/(P1)G<ZU(G/(P1)G). Since N is a unique minimal normal subgroup of G contained in Gp, it follows that N<(P1)G<P1; a contradiction. Thus (P1)G = 1. Hence, P1∩H<ZU (G). If P1∩H≠1, then Gp∩ZU(G) be a non-trivial normal sylow p-subgroup of ZU(G). Now, it follows that Gp∩ZU is supersolvably embedded in G, then GP∩ZU(G) contains a subgroup L of order p is a normal subgroup of G. By the uniqueness and minimality of N, it follows that L = N. Then G∈U, as G/N∈U and ∣N∣ = p; a contradiction, thus P1∩H = 1. Since, P1H is a Hall subgroup of G, it follows that P1<Gp<P1H. Then Gp = P1(Gp∩H) and ∣Gp∩H∣ = p as P1∩H = 1. Also since H is quasinormal subgroup in G, it follows that HK<G, then Gp∩H = Gp∩HK is a normal sylow p-subgroup of HK. It follows that K<NG(Gp∩H). Since Φ(Gp) = 1, it follows that Gp is an elementary abelian, then Gp<NG(Gp∩H). Now, it follows that G = GpK<NG(Gp∩H), i.e., Gp∩H is a normal subgroup of G. By the uniqueness and minimality of N, it follows that Gp∩H = N and so ∣N∣ = ∣Gp∩H∣ = p. Then G∈U, as G/N∈U and ∣N∣ = p; a contradiction.

Case 2: P<Gp

Put π(G) = {p1, p2,…, pn} be a set of all primes dividing the order of G, where p1>p2>…>pn. Since G/P∈U, it follows by Lemma 2.2, that G/P possesses two super solvable subgroups H/P and K/P with ∣G/P:H/P∣ = p1 and ∣G/P:K/P∣ = pn. By Lemma 2.1 (d), the hypothesis carries over H/P and K/P. It follows that by the minimality of G, H and K are in U and ∣G:H∣ = ∣G/P:H/P∣ = p1 and ∣G:K∣ = ∣G/P:K/P∣ = pn. Hence, by Lemma 2.2, G∈U; a final contradiction.

As corollaries of Corollary 3.2 and Theorem 3.3.

Corollary 3.4: Let K be a normal subgroup of a group G such that G/K∈U. If the maximal subgroups of the sylow subgroups of K are Uhq-supplemented in G, then G∈U.

Proof: By Lemma 2.1 (d), the maximal subgroups of the sylow subgroups of K are Uhq-supplemented in K. Then by Corollary 3.2, K possesses an ordered sylow tower. It follows that K has a normal sylow p-subgroup Kp, where p is the largest prime dividing the order of K. Since Kp is characteristic of K and K is a normal subgroup of G, it follows that Kp is a normal subgroup of G. Now consider the factor group G/Kp. Since G/K∈U, it follows that (G/Kp)/(K/Kp)≅G/K∈U and since the maximal subgroups of the sylow subgroups of K are Uhq-supplemented in G, it follows by Lemma 2.1 (c), the maximal subgroups of the sylow subgroups of K/Kp are Uhq-supplemented in G/Kp. Then G/Kp∈U, by the induction on the order of G. Therefore G∈U, by Theorem 3.3.

Corollary 3.5: If the maximal subgroups of the sylow subgroups of a group G are Uhq-supplemented in G, then G∈U.

Proof: By Corollary 3.2, G possesses an ordered sylow tower, then G has a normal sylow p-subgroup Gp, where p is the largest prime dividing the order of G. By Lemma 2.1 (c), our hypothesis carries over G/Gp. Then G/Gp∈U, by the induction on the order of G. Therefore G∈U, by Theorem 3.3.

Some applications: Finally, consider some applications of Theorems 3.1, 3.3 and Corollaries 3.4, 3.5.

Corollary 4.1: Let p be the smallest prime dividing the order of G and let Gp be a sylow p-subgroup of G. If the maximal subgroups of Gp are c-normal in G, then G is p-nilpotent14.

Corollary 4.2: Let K be a normal subgroup of a group G such that G/K∈U. If the maximal subgroups of the sylow subgroups of K are normal in G, then G∈U11.

Corollary 4.3: Let K be a normal subgroup of a group G such that G/K∈U. If the maximal subgroups of the sylow subgroups of K are c-normal in G, then G∈U14.

Corollary 4.4: Let K be a normal subgroup of a group G such that G/K∈U. If the maximal subgroups of the sylow subgroups of K are Uh-normal in G, then G∈U14.

Corollary 4.5: If the maximal subgroups of the sylow subgroups of a group G are normal in G, then G∈U11.

Corollary 4.6: If the maximal subgroups of the sylow subgroups of a group G are c-normal in G, then G∈U3.

Corollary 4.7: If the maximal subgroups of the sylow subgroups of a group G are Uh-normal in G, then G∈U4.

CONCLUSION

This study improves and extends some results of super solvability of the group by using the concept of 𝔍hq-supplemented.

ACKNOWLEDGMENT

The authors gratefully acknowledge the approval and the support of this study from the Deanship of Scientific Research study by the grant No. 8-19-1436-5 K. S. A., Northern Border University, Arar.

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