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.
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.